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Contents Preface xiii List of Acronyms xvii 1 Introduction 1.1 Control System Design Steps . . . . . . . . . . . 1.2...

Author: Petros A. Ioannou Jing Sun

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Contents Preface

xiii

List of Acronyms

xvii

1 Introduction 1.1 Control System Design Steps . . . . . . . . . . . 1.2 Adaptive Control . . . . . . . . . . . . . . . . . . 1.2.1 Robust Control . . . . . . . . . . . . . . . 1.2.2 Gain Scheduling . . . . . . . . . . . . . . 1.2.3 Direct and Indirect Adaptive Control . . 1.2.4 Model Reference Adaptive Control . . . . 1.2.5 Adaptive Pole Placement Control . . . . . 1.2.6 Design of On-Line Parameter Estimators 1.3 A Brief History . . . . . . . . . . . . . . . . . . . 2 Models for Dynamic Systems 2.1 Introduction . . . . . . . . . . . . . . 2.2 State-Space Models . . . . . . . . . 2.2.1 General Description . . . . . 2.2.2 Canonical State-Space Forms 2.3 Input/Output Models . . . . . . . . 2.3.1 Transfer Functions . . . . . . 2.3.2 Coprime Polynomials . . . . 2.4 Plant Parametric Models . . . . . . 2.4.1 Linear Parametric Models . . 2.4.2 Bilinear Parametric Models . v

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26 26 27 27 29 34 34 39 47 49 58

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CONTENTS 2.5

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

3 Stability 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Norms and Lp Spaces . . . . . . . . . . . . . . . . . . 3.2.2 Properties of Functions . . . . . . . . . . . . . . . . . 3.2.3 Positive Definite Matrices . . . . . . . . . . . . . . . . 3.3 Input/Output Stability . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Lp Stability . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The L2δ Norm and I/O Stability . . . . . . . . . . . . 3.3.3 Small Gain Theorem . . . . . . . . . . . . . . . . . . . 3.3.4 Bellman-Gronwall Lemma . . . . . . . . . . . . . . . . 3.4 Lyapunov Stability . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Definition of Stability . . . . . . . . . . . . . . . . . . 3.4.2 Lyapunov’s Direct Method . . . . . . . . . . . . . . . 3.4.3 Lyapunov-Like Functions . . . . . . . . . . . . . . . . 3.4.4 Lyapunov’s Indirect Method . . . . . . . . . . . . . . . 3.4.5 Stability of Linear Systems . . . . . . . . . . . . . . . 3.5 Positive Real Functions and Stability . . . . . . . . . . . . . . 3.5.1 Positive Real and Strictly Positive Real Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 PR and SPR Transfer Function Matrices . . . . . . . 3.6 Stability of LTI Feedback Systems . . . . . . . . . . . . . . . 3.6.1 A General LTI Feedback System . . . . . . . . . . . . 3.6.2 Internal Stability . . . . . . . . . . . . . . . . . . . . . 3.6.3 Sensitivity and Complementary Sensitivity Functions . 3.6.4 Internal Model Principle . . . . . . . . . . . . . . . . . 3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66 66 67 67 72 78 79 79 85 96 101 105 105 108 117 119 120 126 126 132 134 134 135 136 137 139

4 On-Line Parameter Estimation 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 4.2 Simple Examples . . . . . . . . . . . . . . . . . . . 4.2.1 Scalar Example: One Unknown Parameter 4.2.2 First-Order Example: Two Unknowns . . . 4.2.3 Vector Case . . . . . . . . . . . . . . . . . .

144 144 146 146 151 156

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4.3

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4.6 4.7 4.8

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4.2.4 Remarks . . . . . . . . . . . . . . . . . . . . Adaptive Laws with Normalization . . . . . . . . . 4.3.1 Scalar Example . . . . . . . . . . . . . . . . 4.3.2 First-Order Example . . . . . . . . . . . . . 4.3.3 General Plant . . . . . . . . . . . . . . . . . 4.3.4 SPR-Lyapunov Design Approach . . . . . . 4.3.5 Gradient Method . . . . . . . . . . . . . . . 4.3.6 Least-Squares . . . . . . . . . . . . . . . . . 4.3.7 Effect of Initial Conditions . . . . . . . . . Adaptive Laws with Projection . . . . . . . . . . . 4.4.1 Gradient Algorithms with Projection . . . . 4.4.2 Least-Squares with Projection . . . . . . . . Bilinear Parametric Model . . . . . . . . . . . . . . 4.5.1 Known Sign of ρ∗ . . . . . . . . . . . . . . . 4.5.2 Sign of ρ∗ and Lower Bound ρ0 Are Known 4.5.3 Unknown Sign of ρ∗ . . . . . . . . . . . . . Hybrid Adaptive Laws . . . . . . . . . . . . . . . . Summary of Adaptive Laws . . . . . . . . . . . . . Parameter Convergence Proofs . . . . . . . . . . . 4.8.1 Useful Lemmas . . . . . . . . . . . . . . . . 4.8.2 Proof of Corollary 4.3.1 . . . . . . . . . . . 4.8.3 Proof of Theorem 4.3.2 (iii) . . . . . . . . . 4.8.4 Proof of Theorem 4.3.3 (iv) . . . . . . . . . 4.8.5 Proof of Theorem 4.3.4 (iv) . . . . . . . . . 4.8.6 Proof of Corollary 4.3.2 . . . . . . . . . . . 4.8.7 Proof of Theorem 4.5.1(iii) . . . . . . . . . 4.8.8 Proof of Theorem 4.6.1 (iii) . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . .

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161 162 162 165 169 171 180 192 200 203 203 206 208 208 212 215 217 220 220 220 235 236 239 240 241 242 243 245

5 Parameter Identifiers and Adaptive Observers 250 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 5.2 Parameter Identifiers . . . . . . . . . . . . . . . . . . . . . . . 251 5.2.1 Sufficiently Rich Signals . . . . . . . . . . . . . . . . . 252 5.2.2 Parameter Identifiers with Full-State Measurements . 258 5.2.3 Parameter Identifiers with Partial-State Measurements 260 5.3 Adaptive Observers . . . . . . . . . . . . . . . . . . . . . . . . 267

viii

CONTENTS

5.4 5.5

5.6

5.7

5.3.1 The Luenberger Observer . . . . . . . . . . . 5.3.2 The Adaptive Luenberger Observer . . . . . . 5.3.3 Hybrid Adaptive Luenberger Observer . . . . Adaptive Observer with Auxiliary Input . . . . . . Adaptive Observers for Nonminimal Plant Models 5.5.1 Adaptive Observer Based on Realization 1 . . 5.5.2 Adaptive Observer Based on Realization 2 . . Parameter Convergence Proofs . . . . . . . . . . . . 5.6.1 Useful Lemmas . . . . . . . . . . . . . . . . . 5.6.2 Proof of Theorem 5.2.1 . . . . . . . . . . . . 5.6.3 Proof of Theorem 5.2.2 . . . . . . . . . . . . 5.6.4 Proof of Theorem 5.2.3 . . . . . . . . . . . . 5.6.5 Proof of Theorem 5.2.5 . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . .

6 Model Reference Adaptive Control 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . 6.2 Simple Direct MRAC Schemes . . . . . . . . . . 6.2.1 Scalar Example: Adaptive Regulation . . 6.2.2 Scalar Example: Adaptive Tracking . . . 6.2.3 Vector Case: Full-State Measurement . . 6.2.4 Nonlinear Plant . . . . . . . . . . . . . . . 6.3 MRC for SISO Plants . . . . . . . . . . . . . . . 6.3.1 Problem Statement . . . . . . . . . . . . . 6.3.2 MRC Schemes: Known Plant Parameters 6.4 Direct MRAC with Unnormalized Adaptive Laws 6.4.1 Relative Degree n∗ = 1 . . . . . . . . . . 6.4.2 Relative Degree n∗ = 2 . . . . . . . . . . 6.4.3 Relative Degree n∗ = 3 . . . . . . . . . . . 6.5 Direct MRAC with Normalized Adaptive Laws 6.5.1 Example: Adaptive Regulation . . . . . . 6.5.2 Example: Adaptive Tracking . . . . . . . 6.5.3 MRAC for SISO Plants . . . . . . . . . . 6.5.4 Effect of Initial Conditions . . . . . . . . 6.6 Indirect MRAC . . . . . . . . . . . . . . . . . . . 6.6.1 Scalar Example . . . . . . . . . . . . . . .

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267 269 276 279 287 287 292 297 297 301 302 306 309 310

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313 313 315 315 320 325 328 330 331 333 344 345 356 363 373 373 380 384 396 397 398

CONTENTS

6.7

6.8

6.9

ix

6.6.2 Indirect MRAC with Unnormalized Adaptive Laws 6.6.3 Indirect MRAC with Normalized Adaptive Law . . Relaxation of Assumptions in MRAC . . . . . . . . . . . . 6.7.1 Assumption P1: Minimum Phase . . . . . . . . . . 6.7.2 Assumption P2: Upper Bound for the Plant Order 6.7.3 Assumption P3: Known Relative Degree n∗ . . . . 6.7.4 Tunability . . . . . . . . . . . . . . . . . . . . . . . Stability Proofs of MRAC Schemes . . . . . . . . . . . . . 6.8.1 Normalizing Properties of Signal mf . . . . . . . . 6.8.2 Proof of Theorem 6.5.1: Direct MRAC . . . . . . . 6.8.3 Proof of Theorem 6.6.2: Indirect MRAC . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Adaptive Pole Placement Control 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 7.2 Simple APPC Schemes . . . . . . . . . . . . . . . . . 7.2.1 Scalar Example: Adaptive Regulation . . . . 7.2.2 Modified Indirect Adaptive Regulation . . . . 7.2.3 Scalar Example: Adaptive Tracking . . . . . 7.3 PPC: Known Plant Parameters . . . . . . . . . . . . 7.3.1 Problem Statement . . . . . . . . . . . . . . . 7.3.2 Polynomial Approach . . . . . . . . . . . . . 7.3.3 State-Variable Approach . . . . . . . . . . . . 7.3.4 Linear Quadratic Control . . . . . . . . . . . 7.4 Indirect APPC Schemes . . . . . . . . . . . . . . . . 7.4.1 Parametric Model and Adaptive Laws . . . . 7.4.2 APPC Scheme: The Polynomial Approach . . 7.4.3 APPC Schemes: State-Variable Approach . . 7.4.4 Adaptive Linear Quadratic Control (ALQC) 7.5 Hybrid APPC Schemes . . . . . . . . . . . . . . . . 7.6 Stabilizability Issues and Modified APPC . . . . . . 7.6.1 Loss of Stabilizability: A Simple Example . . 7.6.2 Modified APPC Schemes . . . . . . . . . . . 7.6.3 Switched-Excitation Approach . . . . . . . . 7.7 Stability Proofs . . . . . . . . . . . . . . . . . . . . . 7.7.1 Proof of Theorem 7.4.1 . . . . . . . . . . . .

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435 435 437 437 441 443 448 449 450 455 460 467 467 469 479 487 495 499 500 503 507 514 514

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7.8

7.7.2 Proof of Theorem 7.4.2 . . . . . . . . . . . . . . . . . 520 7.7.3 Proof of Theorem 7.5.1 . . . . . . . . . . . . . . . . . 524 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

8 Robust Adaptive Laws 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Plant Uncertainties and Robust Control . . . . . . . . . 8.2.1 Unstructured Uncertainties . . . . . . . . . . . . 8.2.2 Structured Uncertainties: Singular Perturbations 8.2.3 Examples of Uncertainty Representations . . . . 8.2.4 Robust Control . . . . . . . . . . . . . . . . . . . 8.3 Instability Phenomena in Adaptive Systems . . . . . . . 8.3.1 Parameter Drift . . . . . . . . . . . . . . . . . . 8.3.2 High-Gain Instability . . . . . . . . . . . . . . . 8.3.3 Instability Resulting from Fast Adaptation . . . 8.3.4 High-Frequency Instability . . . . . . . . . . . . 8.3.5 Effect of Parameter Variations . . . . . . . . . . 8.4 Modifications for Robustness: Simple Examples . . . . . 8.4.1 Leakage . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Parameter Projection . . . . . . . . . . . . . . . 8.4.3 Dead Zone . . . . . . . . . . . . . . . . . . . . . 8.4.4 Dynamic Normalization . . . . . . . . . . . . . . 8.5 Robust Adaptive Laws . . . . . . . . . . . . . . . . . . . 8.5.1 Parametric Models with Modeling Error . . . . . 8.5.2 SPR-Lyapunov Design Approach with Leakage . 8.5.3 Gradient Algorithms with Leakage . . . . . . . . 8.5.4 Least-Squares with Leakage . . . . . . . . . . . . 8.5.5 Projection . . . . . . . . . . . . . . . . . . . . . . 8.5.6 Dead Zone . . . . . . . . . . . . . . . . . . . . . 8.5.7 Bilinear Parametric Model . . . . . . . . . . . . . 8.5.8 Hybrid Adaptive Laws . . . . . . . . . . . . . . . 8.5.9 Effect of Initial Conditions . . . . . . . . . . . . 8.6 Summary of Robust Adaptive Laws . . . . . . . . . . . 8.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . .

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531 531 532 533 537 540 542 545 546 549 550 552 553 555 557 566 567 572 576 577 583 593 603 604 607 614 617 624 624 626

CONTENTS 9 Robust Adaptive Control Schemes 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Robust Identifiers and Adaptive Observers . . . . . . . . . . 9.2.1 Dominantly Rich Signals . . . . . . . . . . . . . . . . 9.2.2 Robust Parameter Identifiers . . . . . . . . . . . . . 9.2.3 Robust Adaptive Observers . . . . . . . . . . . . . . 9.3 Robust MRAC . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 MRC: Known Plant Parameters . . . . . . . . . . . 9.3.2 Direct MRAC with Unnormalized Adaptive Laws . . 9.3.3 Direct MRAC with Normalized Adaptive Laws . . . 9.3.4 Robust Indirect MRAC . . . . . . . . . . . . . . . . 9.4 Performance Improvement of MRAC . . . . . . . . . . . . . 9.4.1 Modified MRAC with Unnormalized Adaptive Laws 9.4.2 Modified MRAC with Normalized Adaptive Laws . . 9.5 Robust APPC Schemes . . . . . . . . . . . . . . . . . . . . 9.5.1 PPC: Known Parameters . . . . . . . . . . . . . . . 9.5.2 Robust Adaptive Laws for APPC Schemes . . . . . . 9.5.3 Robust APPC: Polynomial Approach . . . . . . . . 9.5.4 Robust APPC: State Feedback Law . . . . . . . . . 9.5.5 Robust LQ Adaptive Control . . . . . . . . . . . . . 9.6 Adaptive Control of LTV Plants . . . . . . . . . . . . . . . 9.7 Adaptive Control for Multivariable Plants . . . . . . . . . . 9.7.1 Decentralized Adaptive Control . . . . . . . . . . . . 9.7.2 The Command Generator Tracker Approach . . . . 9.7.3 Multivariable MRAC . . . . . . . . . . . . . . . . . . 9.8 Stability Proofs of Robust MRAC Schemes . . . . . . . . . 9.8.1 Properties of Fictitious Normalizing Signal . . . . . 9.8.2 Proof of Theorem 9.3.2 . . . . . . . . . . . . . . . . 9.9 Stability Proofs of Robust APPC Schemes . . . . . . . . . . 9.9.1 Proof of Theorem 9.5.2 . . . . . . . . . . . . . . . . 9.9.2 Proof of Theorem 9.5.3 . . . . . . . . . . . . . . . . 9.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Swapping Lemmas . . . . . . . . . . . . . . . . . . . . . . . B Optimization Techniques . . . . . . . . . . . . . . . . . . . . B.1 Notation and Mathematical Background . . . . . . . B.2 The Method of Steepest Descent (Gradient Method)

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635 635 636 639 644 649 651 652 657 667 688 694 698 704 710 711 714 716 723 731 733 735 736 737 740 745 745 749 760 760 764 769 775 784 784 786

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Newton’s Method . . . . . . . . . . . . . . . . . . . . . 787 Gradient Projection Method . . . . . . . . . . . . . . 789 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 792

Bibliography

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Index

819

License Agreement and Limited Warranty

822

Preface The area of adaptive control has grown to be one of the richest in terms of algorithms, design techniques, analytical tools, and modifications. Several books and research monographs already exist on the topics of parameter estimation and adaptive control. Despite this rich literature, the field of adaptive control may easily appear to an outsider as a collection of unrelated tricks and modifications. Students are often overwhelmed and sometimes confused by the vast number of what appear to be unrelated designs and analytical methods achieving similar results. Researchers concentrating on different approaches in adaptive control often find it difficult to relate their techniques with others without additional research efforts. The purpose of this book is to alleviate some of the confusion and difficulty in understanding the design, analysis, and robustness of a wide class of adaptive control for continuous-time plants. The book is the outcome of several years of research, whose main purpose was not to generate new results, but rather unify, simplify, and present in a tutorial manner most of the existing techniques for designing and analyzing adaptive control systems. The book is written in a self-contained fashion to be used as a textbook on adaptive systems at the senior undergraduate, or first and second graduate level. It is assumed that the reader is familiar with the materials taught in undergraduate courses on linear systems, differential equations, and automatic control. The book is also useful for an industrial audience where the interest is to implement adaptive control rather than analyze its stability properties. Tables with descriptions of adaptive control schemes presented in the book are meant to serve this audience. The personal computer floppy disk, included with the book, provides several examples of simple adaptive xiii

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control systems that will help the reader understand some of the implementation aspects of adaptive systems. A significant part of the book, devoted to parameter estimation and learning in general, provides techniques and algorithms for on-line fitting of dynamic or static models to data generated by real systems. The tools for design and analysis presented in the book are very valuable in understanding and analyzing similar parameter estimation problems that appear in neural networks, fuzzy systems, and other universal approximators. The book will be of great interest to the neural and fuzzy logic audience who will benefit from the strong similarity that exists between adaptive systems, whose stability properties are well established, and neural networks, fuzzy logic systems where stability and convergence issues are yet to be resolved. The book is organized as follows: Chapter 1 is used to introduce adaptive control as a method for controlling plants with parametric uncertainty. It also provides some background and a brief history of the development of adaptive control. Chapter 2 presents a review of various plant model representations that are useful for parameter identification and control. A considerable number of stability results that are useful in analyzing and understanding the properties of adaptive and nonlinear systems in general are presented in Chapter 3. Chapter 4 deals with the design and analysis of online parameter estimators or adaptive laws that form the backbone of every adaptive control scheme presented in the chapters to follow. The design of parameter identifiers and adaptive observers for stable plants is presented in Chapter 5. Chapter 6 is devoted to the design and analysis of a wide class of model reference adaptive controllers for minimum phase plants. The design of adaptive control for plants that are not necessarily minimum phase is presented in Chapter 7. These schemes are based on pole placement control strategies and are referred to as adaptive pole placement control. While Chapters 4 through 7 deal with plant models that are free of disturbances, unmodeled dynamics and noise, Chapters 8 and 9 deal with the robustness issues in adaptive control when plant model uncertainties, such as bounded disturbances and unmodeled dynamics, are present. The book can be used in various ways. The reader who is familiar with stability and linear systems may start from Chapter 4. An introductory course in adaptive control could be covered in Chapters 1, 2, and 4 to 9, by excluding the more elaborate and difficult proofs of theorems that are

PREFACE

xv

presented either in the last section of chapters or in the appendices. Chapter 3 could be used for reference and for covering relevant stability results that arise during the course. A higher-level course intended for graduate students that are interested in a deeper understanding of adaptive control could cover all chapters with more emphasis on the design and stability proofs. A course for an industrial audience could contain Chapters 1, 2, and 4 to 9 with emphasis on the design of adaptive control algorithms rather than stability proofs and convergence.

Acknowledgments The writing of this book has been surprisingly difficult and took a long time to evolve to its present form. Several versions of the book were completed only to be put aside after realizing that new results and techniques would lead to a better version. In the meantime, both of us started our families that soon enough expanded. If it were not for our families, we probably could have finished the book a year or two earlier. Their love and company, however, served as an insurance that we would finish it one day. A long list of friends and colleagues have helped us in the preparation of the book in many different ways. We are especially grateful to Petar Kokotovi´c who introduced the first author to the field of adaptive control back in 1979. Since then he has been a great advisor, friend, and colleague. His continuous enthusiasm and hard work for research has been the strongest driving force behind our research and that of our students. We thank Brian Anderson, Karl ˚ Astr¨om, Mike Athans, Bo Egardt, Graham Goodwin, Rick Johnson, Gerhard Kreisselmeier, Yoan Landau, Lennart Ljung, David Mayne, late R. Monopoli, Bob Narendra, and Steve Morse for their work, interactions, and continuous enthusiasm in adaptive control that helped us lay the foundations of most parts of the book. We would especially like to express our deepest appreciation to Laurent Praly and Kostas Tsakalis. Laurent was the first researcher to recognize and publicize the beneficial effects of dynamic normalization on robustness that opened the way to a wide class of robust adaptive control algorithms addressed in the book. His interactions with us and our students is highly appreciated. Kostas, a former student of the first author, is responsible for many mathematical tools and stability arguments used in Chapters 6 and

xvi

PREFACE

9. His continuous interactions helped us to decipher many of the cryptic concepts and robustness properties of model reference adaptive control. We are thankful to our former and current students and visitors who collaborated with us in research and contributed to this work: Farid AhmedZaid, C. C. Chien, Aniruddha Datta, Marios Polycarpou, Houmair Raza, Alex Stotsky, Tim Sun, Hualin Tan, Gang Tao, Hui Wang, Tom Xu, and Youping Zhang. We are grateful to many colleagues for stimulating discussions at conferences, workshops, and meetings. They have helped us broaden our understanding of the field. In particular, we would like to mention Anu Annaswamy, Erwei Bai, Bob Bitmead, Marc Bodson, Stephen Boyd, Sara Dasgupta, the late Howard Elliot, Li-chen Fu, Fouad Giri, David Hill, Ioannis Kanellakopoulos, Pramod Khargonekar, Hassan Khalil, Bob Kosut, Jim Krause, Miroslav Krsti´c, Rogelio Lozano-Leal, Iven Mareels, Rick Middleton, David Mudget, Romeo Ortega, Brad Riedle, Charles Rohrs, Ali Saberi, Shankar Sastry, Lena Valavani, Jim Winkelman, and Erik Ydstie. We would also like to extend our thanks to our colleagues at the University of Southern California, Wayne State University, and Ford Research Laboratory for their friendship, support, and technical interactions. Special thanks, on behalf of the second author, go to the members of the Control Systems Department of Ford Research Laboratory, and Jessy Grizzle and Anna Stefanopoulou of the University of Michigan. Finally, we acknowledge the support of several organizations including Ford Motor Company, General Motors Project Trilby, National Science Foundation, Rockwell International, and Lockheed. Special thanks are due to Bob Borcherts, Roger Fruechte, Neil Schilke, and James Rillings of former Project Trilby; Bill Powers, Mike Shulman, and Steve Eckert of Ford Motor Company; and Bob Rooney and Houssein Youseff of Lockheed whose support of our research made this book possible. Petros A. Ioannou Jing Sun

List of Acronyms ALQC APPC B-G BIBO CEC I/O LKY LQ LTI LTV MIMO MKY MRAC MRC PE PI PPC PR SISO SPR TV UCO a.s. e.s. m.s.s. u.a.s. u.b. u.s. u.u.b. w.r.t.

Adaptive linear quadratic control Adaptive pole placement control Bellman Gronwall (lemma) Bounded-input bounded-output Certainty equivalence control Input/output Lefschetz-Kalman-Yakubovich (lemma) Linear quadratic Linear time invariant Linear time varying Multi-input multi-output Meyer-Kalman-Yakubovich (lemma) Model reference adaptive control Model reference control Persistently exciting Proportional plus integral Pole placement control Positive real Single input single output Strictly positive real Time varying Uniformly completely observable Asymptotically stable Exponentially stable (In the) mean square sense Uniformly asymptotically stable Uniformly bounded Uniformly stable Uniformly ultimately bounded With respect to

xvii

18

PREFACE

Chapter 1

Introduction 1.1

Control System Design Steps

The design of a controller that can alter or modify the behavior and response of an unknown plant to meet certain performance requirements can be a tedious and challenging problem in many control applications. By plant, we mean any process characterized by a certain number of inputs u and outputs y, as shown in Figure 1.1. The plant inputs u are processed to produce several plant outputs y that represent the measured output response of the plant. The control design task is to choose the input u so that the output response y(t) satisfies certain given performance requirements. Because the plant process is usually complex, i.e., it may consist of various mechanical, electronic, hydraulic parts, etc., the appropriate choice of u is in general not straightforward. The control design steps often followed by most control engineers in choosing the input u are shown in Figure 1.2 and are explained below.

Inputs u

H ©

Plant Process P

Figure 1.1 Plant representation. 1

H Outputs © y

2

CHAPTER 1. INTRODUCTION

Step 1. Modeling The task of the control engineer in this step is to understand the processing mechanism of the plant, which takes a given input signal u(t) and produces the output response y(t), to the point that he or she can describe it in the form of some mathematical equations. These equations constitute the mathematical model of the plant. An exact plant model should produce the same output response as the plant, provided the input to the model and initial conditions are exactly the same as those of the plant. The complexity of most physical plants, however, makes the development of such an exact model unwarranted or even impossible. But even if the exact plant model becomes available, its dimension is likely to be infinite, and its description nonlinear or time varying to the point that its usefulness from the control design viewpoint is minimal or none. This makes the task of modeling even more difficult and challenging, because the control engineer has to come up with a mathematical model that describes accurately the input/output behavior of the plant and yet is simple enough to be used for control design purposes. A simple model usually leads to a simple controller that is easier to understand and implement, and often more reliable for practical purposes. A plant model may be developed by using physical laws or by processing the plant input/output (I/O) data obtained by performing various experiments. Such a model, however, may still be complicated enough from the control design viewpoint and further simplifications may be necessary. Some of the approaches often used to obtain a simplified model are (i) (ii)

Linearization around operating points Model order reduction techniques

In approach (i) the plant is approximated by a linear model that is valid around a given operating point. Different operating points may lead to several different linear models that are used as plant models. Linearization is achieved by using Taylor’s series expansion and approximation, fitting of experimental data to a linear model, etc. In approach (ii) small effects and phenomena outside the frequency range of interest are neglected leading to a lower order and simpler plant model. The reader is referred to references [67, 106] for more details on model reduction techniques and approximations.

1.1. CONTROL SYSTEM DESIGN STEPS

u

Plant P

H ©

3

H ©y

Step 1: Modeling

?

u

Plant Model H © P m

H © yˆ

Step 2: Controller Design

?

Input Command

Uncertainty H © ∆ H Controller © H C ©

u

Plant Model H © P m

A¢ H © Σl

yˆ H ©

Step 3: Implementation

Input Command

? H © Controller H C ©

u

H ©

Plant P

y H ©

Figure 1.2 Control system design steps. In general, the task of modeling involves a good understanding of the plant process and performance requirements, and may require some experience from the part of the control engineer. Step 2. Controller Design Once a model of the plant is available, one can proceed with the controller design. The controller is designed to meet the performance requirements for

4

CHAPTER 1. INTRODUCTION

the plant model. If the model is a good approximation of the plant, then one would hope that the controller performance for the plant model would be close to that achieved when the same controller is applied to the plant. Because the plant model is always an approximation of the plant, the effect of any discrepancy between the plant and the model on the performance of the controller will not be known until the controller is applied to the plant in Step 3. One, however, can take an intermediate step and analyze the properties of the designed controller for a plant model that includes a class of plant model uncertainties denoted by 4 that are likely to appear in the plant. If 4 represents most of the unmodeled plant phenomena, its representation in terms of mathematical equations is not possible. Its characterization, however, in terms of some known bounds may be possible in many applications. By considering the existence of a general class of uncertainties 4 that are likely to be present in the plant, the control engineer may be able to modify or redesign the controller to be less sensitive to uncertainties, i.e., to be more robust with respect to 4. This robustness analysis and redesign improves the potential for a successful implementation in Step 3. Step 3. Implementation In this step, a controller designed in Step 2, which is shown to meet the performance requirements for the plant model and is robust with respect to possible plant model uncertainties 4, is ready to be applied to the unknown plant. The implementation can be done using a digital computer, even though in some applications analog computers may be used too. Issues, such as the type of computer available, the type of interface devices between the computer and the plant, software tools, etc., need to be considered a priori. Computer speed and accuracy limitations may put constraints on the complexity of the controller that may force the control engineer to go back to Step 2 or even Step 1 to come up with a simpler controller without violating the performance requirements. Another important aspect of implementation is the final adjustment, or as often called the tuning, of the controller to improve performance by compensating for the plant model uncertainties that are not accounted for during the design process. Tuning is often done by trial and error, and depends very much on the experience and intuition of the control engineer. In this book we will concentrate on Step 2. We will be dealing with

1.2. ADAPTIVE CONTROL

5

the design of control algorithms for a class of plant models described by the linear differential equation x˙ = Ax + Bu, x(0) = x0 y = C > x + Du

(1.1.1)

In (1.1.1) x ∈ Rn is the state of the model, u ∈ Rr the plant input, and y ∈ Rl the plant model output. The matrices A ∈ Rn×n , B ∈ Rn×r , C ∈ Rn×l , and D ∈ Rl×r could be constant or time varying. This class of plant models is quite general because it can serve as an approximation of nonlinear plants around operating points. A controller based on the linear model (1.1.1) is expected to be simpler and easier to understand than a controller based on a possibly more accurate but nonlinear plant model. The class of plant models given by (1.1.1) can be generalized further if we allow the elements of A, B, and C to be completely unknown and changing with time or operating conditions. The control of plant models (1.1.1) with A, B, C, and D unknown or partially known is covered under the area of adaptive systems and is the main topic of this book.

1.2

Adaptive Control

According to Webster’s dictionary, to adapt means “to change (oneself) so that one’s behavior will conform to new or changed circumstances.” The words “adaptive systems” and “adaptive control” have been used as early as 1950 [10, 27]. The design of autopilots for high-performance aircraft was one of the primary motivations for active research on adaptive control in the early 1950s. Aircraft operate over a wide range of speeds and altitudes, and their dynamics are nonlinear and conceptually time varying. For a given operating point, specified by the aircraft speed (Mach number) and altitude, the complex aircraft dynamics can be approximated by a linear model of the same form as (1.1.1). For example, for an operating point i, the linear aircraft model has the following form [140]: x˙ = Ai x + Bi u, x(0) = x0 y = Ci> x + Di u

(1.2.1)

where Ai , Bi , Ci , and Di are functions of the operating point i. As the aircraft goes through different flight conditions, the operating point changes

6

CHAPTER 1. INTRODUCTION

¢¢¸ -

Input - Controller Command

u -

Plant P

-y

¢¢

¢

Strategy for ¾ Adjusting ¾ Controller Gains

u(t) y(t)

Figure 1.3 Controller structure with adjustable controller gains. leading to different values for Ai , Bi , Ci , and Di . Because the output response y(t) carries information about the state x as well as the parameters, one may argue that in principle, a sophisticated feedback controller should be able to learn about parameter changes by processing y(t) and use the appropriate gains to accommodate them. This argument led to a feedback control structure on which adaptive control is based. The controller structure consists of a feedback loop and a controller with adjustable gains as shown in Figure 1.3. The way of changing the controller gains in response to changes in the plant and disturbance dynamics distinguishes one scheme from another.

1.2.1

Robust Control

A constant gain feedback controller may be designed to cope with parameter changes provided that such changes are within certain bounds. A block diagram of such a controller is shown in Figure 1.4 where G(s) is the transfer function of the plant and C(s) is the transfer function of the controller. The transfer function from y ∗ to y is y C(s)G(s) = ∗ y 1 + C(s)G(s)

(1.2.2)

where C(s) is to be chosen so that the closed-loop plant is stable, despite parameter changes or uncertainties in G(s), and y ≈ y ∗ within the frequency range of interest. This latter condition can be achieved if we choose C(s)

1.2. ADAPTIVE CONTROL y ∗ +- l Σ −6

- C(s)

7 u - Plant G(s)

y-

Figure 1.4 Constant gain feedback controller. so that the loop gain |C(jw)G(jw)| is as large as possible in the frequency spectrum of y ∗ provided, of course, that large loop gain does not violate closed-loop stability requirements. The tracking and stability objectives can be achieved through the design of C(s) provided the changes within G(s) are within certain bounds. More details about robust control will be given in Chapter 8. Robust control is not considered to be an adaptive system even though it can handle certain classes of parametric and dynamic uncertainties.

1.2.2

Gain Scheduling

Let us consider the aircraft model (1.2.1) where for each operating point i, i = 1, 2, . . . , N , the parameters Ai , Bi , Ci , and Di are known. For a given operating point i, a feedback controller with constant gains, say θi , can be designed to meet the performance requirements for the corresponding linear model. This leads to a controller, say C(θ), with a set of gains {θ1 , θ2 , ..., θi , ..., θN } covering N operating points. Once the operating point, say i, is detected the controller gains can be changed to the appropriate value of θi obtained from the precomputed gain set. Transitions between different operating points that lead to significant parameter changes may be handled by interpolation or by increasing the number of operating points. The two elements that are essential in implementing this approach is a look-up table to store the values of θi and the plant auxiliary measurements that correlate well with changes in the operating points. The approach is called gain scheduling and is illustrated in Figure 1.5. The gain scheduler consists of a look-up table and the appropriate logic for detecting the operating point and choosing the corresponding value of θi from the table. In the case of aircraft, the auxiliary measurements are the Mach number and the dynamic pressure. With this approach plant

8

CHAPTER 1. INTRODUCTION Auxiliary Gain ¾Measurements θi ¡ Scheduler

¡ Command or ¡ Reference Signal - Controller -

C(θ)

u

- Plant

y-

¡ ¡ ª

Figure 1.5 Gain scheduling. parameter variations can be compensated by changing the controller gains as functions of the auxiliary measurements. The advantage of gain scheduling is that the controller gains can be changed as quickly as the auxiliary measurements respond to parameter changes. Frequent and rapid changes of the controller gains, however, may lead to instability [226]; therefore, there is a limit as to how often and how fast the controller gains can be changed. One of the disadvantages of gain scheduling is that the adjustment mechanism of the controller gains is precomputed off-line and, therefore, provides no feedback to compensate for incorrect schedules. Unpredictable changes in the plant dynamics may lead to deterioration of performance or even to complete failure. Another possible drawback of gain scheduling is the high design and implementation costs that increase with the number of operating points. Despite its limitations, gain scheduling is a popular method for handling parameter variations in flight control [140, 210] and other systems [8].

1.2.3

Direct and Indirect Adaptive Control

An adaptive controller is formed by combining an on-line parameter estimator, which provides estimates of unknown parameters at each instant, with a control law that is motivated from the known parameter case. The way the parameter estimator, also referred to as adaptive law in the book, is combined with the control law gives rise to two different approaches. In the first approach, referred to as indirect adaptive control, the plant parameters are estimated on-line and used to calculate the controller parameters. This

1.2. ADAPTIVE CONTROL

9

approach has also been referred to as explicit adaptive control, because the design is based on an explicit plant model. In the second approach, referred to as direct adaptive control, the plant model is parameterized in terms of the controller parameters that are estimated directly without intermediate calculations involving plant parameter estimates. This approach has also been referred to as implicit adaptive control because the design is based on the estimation of an implicit plant model. In indirect adaptive control, the plant model P (θ∗ ) is parameterized with respect to some unknown parameter vector θ∗ . For example, for a linear time invariant (LTI) single-input single-output (SISO) plant model, θ∗ may represent the unknown coefficients of the numerator and denominator of the plant model transfer function. An on-line parameter estimator generates an estimate θ(t) of θ∗ at each time t by processing the plant input u and output y. The parameter estimate θ(t) specifies an estimated plant model characterized by Pˆ (θ(t)) that for control design purposes is treated as the “true” plant model and is used to calculate the controller parameter or gain vector θc (t) by solving a certain algebraic equation θc (t) = F (θ(t)) at each time t. The form of the control law C(θc ) and algebraic equation θc = F (θ) is chosen to be the same as that of the control law C(θc∗ ) and equation θc∗ = F (θ∗ ) that could be used to meet the performance requirements for the plant model P (θ∗ ) if θ∗ was known. It is, therefore, clear that with this approach, C(θc (t)) is designed at each time t to satisfy the performance requirements for the estimated plant model Pˆ (θ(t)), which may be different from the unknown plant model P (θ∗ ). Therefore, the principal problem in indirect adaptive control is to choose the class of control laws C(θc ) and the class of parameter estimators that generate θ(t) as well as the algebraic equation θc (t) = F (θ(t)) so that C(θc (t)) meets the performance requirements for the plant model P (θ∗ ) with unknown θ∗ . We will study this problem in great detail in Chapters 6 and 7, and consider the robustness properties of indirect adaptive control in Chapters 8 and 9. The block diagram of an indirect adaptive control scheme is shown in Figure 1.6. In direct adaptive control, the plant model P (θ∗ ) is parameterized in terms of the unknown controller parameter vector θc∗ , for which C(θc∗ ) meets the performance requirements, to obtain the plant model Pc (θc∗ ) with exactly the same input/output characteristics as P (θ∗ ). The on-line parameter estimator is designed based on Pc (θc∗ ) instead of

10

CHAPTER 1. INTRODUCTION

µ ¡ - Controller - C(θc )

Input ¡ Command r

u -

Plant P (θ∗ )

y

-

?

On-Line ¾ Parameter ¾r Estimation of θ∗ θ(t) ? θc

Calculations θc (t) = F (θ(t))

Figure 1.6 Indirect adaptive control. P (θ∗ ) to provide direct estimates θc (t) of θc∗ at each time t by processing the plant input u and output y. The estimate θc (t) is then used to update the controller parameter vector θc without intermediate calculations. The choice of the class of control laws C(θc ) and parameter estimators generating θc (t) for which C(θc (t)) meets the performance requirements for the plant model P (θ∗ ) is the fundamental problem in direct adaptive control. The properties of the plant model P (θ∗ ) are crucial in obtaining the parameterized plant model Pc (θc∗ ) that is convenient for on-line estimation. As a result, direct adaptive control is restricted to a certain class of plant models. As we will show in Chapter 6, a class of plant models that is suitable for direct adaptive control consists of all SISO LTI plant models that are minimum-phase, i.e., their zeros are located in Re [s] < 0. The block diagram of direct adaptive control is shown in Figure 1.7. The principle behind the design of direct and indirect adaptive control shown in Figures 1.6 and 1.7 is conceptually simple. The design of C(θc ) treats the estimates θc (t) (in the case of direct adaptive control) or the estimates θ(t) (in the case of indirect adaptive control) as if they were the true parameters. This design approach is called certainty equivalence and can be used to generate a wide class of adaptive control schemes by combining different on-line parameter estimators with different control laws.

1.2. ADAPTIVE CONTROL

µ ¡ - Controller - C(θc )

11

u -

Input ¡ Command r

-

Plant P (θ∗ ) → Pc (θc∗ )

y

¾ On-Line Parameter Estimation of θc∗ ¾

r

-

θc Figure 1.7 Direct adaptive control. The idea behind the certainty equivalence approach is that as the parameter estimates θc (t) and θ(t) converge to the true ones θc∗ and θ∗ , respectively, the performance of the adaptive controller C(θc ) tends to that achieved by C(θc∗ ) in the case of known parameters. The distinction between direct and indirect adaptive control may be confusing to most readers for the following reasons: The direct adaptive control structure shown in Figure 1.7 can be made identical to that of the indirect adaptive control by including a block for calculations with an identity transformation between updated parameters and controller parameters. In general, for a given plant model the distinction between the direct and indirect approach becomes clear if we go into the details of design and analysis. For example, direct adaptive control can be shown to meet the performance requirements, which involve stability and asymptotic tracking, for a minimum-phase plant. It is still not clear how to design direct schemes for nonminimum-phase plants. The difficulty arises from the fact that, in general, a convenient (for the purpose of estimation) parameterization of the plant model in terms of the desired controller parameters is not possible for nonminimum-phase plant models. Indirect adaptive control, on the other hand, is applicable to both minimum- and nonminimum-phase plants. In general, however, the mapping 4 between θ(t) and θc (t), defined by the algebraic equation θc (t) = F (θ(t)), cannot be guaranteed to exist at each time t giving rise to the so-called stabilizability problem that is discussed in Chapter 7. As we will show in

12

CHAPTER 1. INTRODUCTION

-

Reference Model Wm (s)

y − ?m e1 Σn +6

r- Controller C(θc∗ )

u - Plant G(s)

y

Figure 1.8 Model reference control. Chapter 7, solutions to the stabilizability problem are possible at the expense of additional complexity. Efforts to relax the minimum-phase assumption in direct adaptive control and resolve the stabilizability problem in indirect adaptive control led to adaptive control schemes where both the controller and plant parameters are estimated on-line, leading to combined direct/indirect schemes that are usually more complex [112].

1.2.4

Model Reference Adaptive Control

Model reference adaptive control (MRAC) is derived from the model following problem or model reference control (MRC) problem. In MRC, a good understanding of the plant and the performance requirements it has to meet allow the designer to come up with a model, referred to as the reference model, that describes the desired I/O properties of the closed-loop plant. The objective of MRC is to find the feedback control law that changes the structure and dynamics of the plant so that its I/O properties are exactly the same as those of the reference model. The structure of an MRC scheme for a LTI, SISO plant is shown in Figure 1.8. The transfer function Wm (s) of the reference model is designed so that for a given reference input signal r(t) the output ym (t) of the reference model represents the desired response the plant output y(t) should follow. The feedback controller denoted by C(θc∗ ) is designed so that all signals are bounded and the closed-loop plant transfer function from r to y is equal to Wm (s). This transfer function matching guarantees that for any given reference input r(t), the tracking error

1.2. ADAPTIVE CONTROL

13

- Reference Model

ym

Wm (s)

µ ¡ - Controller - C(θc )

Input ¡ Command r

− ?e Σl1+ 6 u -

Plant P (θ∗ )

y

?

On-Line ¾ Parameter ¾r Estimation of θ∗ θ(t) ? θc

Calculations θc (t) = F (θ(t))

Figure 1.9 Indirect MRAC. 4

e1 = y − ym , which represents the deviation of the plant output from the desired trajectory ym , converges to zero with time. The transfer function matching is achieved by canceling the zeros of the plant transfer function G(s) and replacing them with those of Wm (s) through the use of the feedback controller C(θc∗ ). The cancellation of the plant zeros puts a restriction on the plant to be minimum phase, i.e., have stable zeros. If any plant zero is unstable, its cancellation may easily lead to unbounded signals. The design of C(θc∗ ) requires the knowledge of the coefficients of the plant transfer function G(s). If θ∗ is a vector containing all the coefficients of G(s) = G(s, θ∗ ), then the parameter vector θc∗ may be computed by solving an algebraic equation of the form θc∗ = F (θ∗ )

(1.2.3)

It is, therefore, clear that for the MRC objective to be achieved the plant model has to be minimum phase and its parameter vector θ∗ has to be known exactly.

14

CHAPTER 1. INTRODUCTION

- Reference Model

ym

Wm (s)

−e ? 1 Σl +6

µ ¡ - Controller - C(θ )

u -

c

Input Command ¡ r

-

Plant P (θ∗ ) → Pc (θc∗ ) ¾ On-Line Parameter Estimation of θc∗ ¾

•

y

r

θc Figure 1.10 Direct MRAC. When θ∗ is unknown the MRC scheme of Figure 1.8 cannot be implemented because θc∗ cannot be calculated using (1.2.3) and is, therefore, unknown. One way of dealing with the unknown parameter case is to use the certainty equivalence approach to replace the unknown θc∗ in the control law with its estimate θc (t) obtained using the direct or the indirect approach. The resulting control schemes are known as MRAC and can be classified as indirect MRAC shown in Figure 1.9 and direct MRAC shown in Figure 1.10. Different choices of on-line parameter estimators lead to further classifications of MRAC. These classifications and the stability properties of both direct and indirect MRAC will be studied in detail in Chapter 6. Other approaches similar to the certainty equivalence approach may be used to design direct and indirect MRAC schemes. The structure of these schemes is a modification of those in Figures 1.9 and 1.10 and will be studied in Chapter 6.

1.2.5

Adaptive Pole Placement Control

Adaptive pole placement control (APPC) is derived from the pole placement control (PPC) and regulation problems used in the case of LTI plants with known parameters.

1.2. ADAPTIVE CONTROL

Input Command r

15

- Controller - C(θ ∗ ) c

- Plant

G(s)

y-

Figure 1.11 Pole placement control. In PPC, the performance requirements are translated into desired locations of the poles of the closed-loop plant. A feedback control law is then developed that places the poles of the closed-loop plant at the desired locations. A typical structure of a PPC scheme for a LTI, SISO plant is shown in Figure 1.11. The structure of the controller C(θc∗ ) and the parameter vector θc∗ are chosen so that the poles of the closed-loop plant transfer function from r to y are equal to the desired ones. The vector θc∗ is usually calculated using an algebraic equation of the form θc∗ = F (θ∗ )

(1.2.4)

where θ∗ is a vector with the coefficients of the plant transfer function G(s). If θ∗ is known, then θc∗ is calculated from (1.2.4) and used in the control law. When θ∗ is unknown, θc∗ is also unknown, and the PPC scheme of Figure 1.11 cannot be implemented. As in the case of MRC, we can deal with the unknown parameter case by using the certainty equivalence approach to replace the unknown vector θc∗ with its estimate θc (t). The resulting scheme is referred to as adaptive pole placement control (APPC). If θc (t) is updated directly using an on-line parameter estimator, the scheme is referred to as direct APPC. If θc (t) is calculated using the equation θc (t) = F (θ(t))

(1.2.5)

where θ(t) is the estimate of θ∗ generated by an on-line estimator, the scheme is referred to as indirect APPC. The structure of direct and indirect APPC is the same as that shown in Figures 1.6 and 1.7 respectively for the general case. The design of APPC schemes is very flexible with respect to the choice of the form of the controller C(θc ) and of the on-line parameter estimator.

16

CHAPTER 1. INTRODUCTION

For example, the control law may be based on the linear quadratic design technique, frequency domain design techniques, or any other PPC method used in the known parameter case. Various combinations of on-line estimators and control laws lead to a wide class of APPC schemes that are studied in detail in Chapter 7. APPC schemes are often referred to as self-tuning regulators in the literature of adaptive control and are distinguished from MRAC. The distinction between APPC and MRAC is more historical than conceptual because as we will show in Chapter 7, MRAC can be considered as a special class of APPC. MRAC was first developed for continuous-time plants for model following, whereas APPC was initially developed for discrete-time plants in a stochastic environment using minimization techniques.

1.2.6

Design of On-Line Parameter Estimators

As we mentioned in the previous sections, an adaptive controller may be considered as a combination of an on-line parameter estimator with a control law that is derived from the known parameter case. The way this combination occurs and the type of estimator and control law used gives rise to a wide class of different adaptive controllers with different properties. In the literature of adaptive control the on-line parameter estimator has often been referred to as the adaptive law, update law, or adjustment mechanism. In this book we will often refer to it as the adaptive law. The design of the adaptive law is crucial for the stability properties of the adaptive controller. As we will see in this book the adaptive law introduces a multiplicative nonlinearity that makes the closed-loop plant nonlinear and often time varying. Because of this, the analysis and understanding of the stability and robustness of adaptive control schemes are more challenging. Some of the basic methods used to design adaptive laws are (i) Sensitivity methods (ii) Positivity and Lyapunov design (iii) Gradient method and least-squares methods based on estimation error cost criteria The last three methods are used in Chapters 4 and 8 to design a wide class of adaptive laws. The sensitivity method is one of the oldest methods used in the design of adaptive laws and will be briefly explained in this section

1.2. ADAPTIVE CONTROL

17

together with the other three methods for the sake of completeness. It will not be used elsewhere in this book for the simple reason that in theory the adaptive laws based on the last three methods can be shown to have better stability properties than those based on the sensitivity method. (i) Sensitivity methods This method became very popular in the 1960s [34, 104], and it is still used in many industrial applications for controlling plants with uncertainties. In adaptive control, the sensitivity method is used to design the adaptive law so that the estimated parameters are adjusted in a direction that minimizes a certain performance function. The adaptive law is driven by the partial derivative of the performance function with respect to the estimated parameters multiplied by an error signal that characterizes the mismatch between the actual and desired behavior. This derivative is called sensitivity function and if it can be generated on-line then the adaptive law is implementable. In most formulations of adaptive control, the sensitivity function cannot be generated on-line, and this constitutes one of the main drawbacks of the method. The use of approximate sensitivity functions that are implementable leads to adaptive control schemes whose stability properties are either weak or cannot be established. As an example let us consider the design of an adaptive law for updating the controller parameter vector θc of the direct MRAC scheme of Figure 1.10. The tracking error e1 represents the deviation of the plant output y from 4 that of the reference model, i.e., e1 = y − ym . Because θc = θc∗ implies that e1 = 0 at steady state, a nonzero value of e1 may be taken to imply that θc 6= θc∗ . Because y depends on θc , i.e., y = y(θc ) we have e1 = e1 (θc ) and, therefore, one way of reducing e1 to zero is to adjust θc in a direction that minimizes a certain cost function of e1 . A simple cost function for e1 is the quadratic function J(θc ) =

e21 (θc ) 2

(1.2.6)

A simple method for adjusting θc to minimize J(θc ) is the method of steepest descent or gradient method (see Appendix B) that gives us the adaptive law θ˙c = −γ∇J(θc ) = −γe1 ∇e1 (θc )

(1.2.7)

18

CHAPTER 1. INTRODUCTION

where

·

∂e1 ∂e1 ∂e1 ∇e1 (θc ) = , , ..., ∂θc1 ∂θc2 ∂θcn is the gradient of e1 with respect to 4

¸>

(1.2.8)

θc = [θc1 , θc2 , ..., θcn ]> Because ∇e1 (θc ) = ∇y(θc ) we have θ˙c = −γe1 ∇y(θc )

(1.2.9)

where γ > 0 is an arbitrary design constant referred to as the adaptive gain ∂y and ∂θ , i = 1, 2, ..., n are the sensitivity functions of y with respect to the ci elements of the controller parameter vector θc . The sensitivity functions ∂y ∂θci represent the sensitivity of the plant output to changes in the controller parameter θc . In (1.2.7) the parameter vector θc is adjusted in the direction of steepest e2 (θ ) descent that decreases J(θc ) = 1 2 c . If J(θc ) is a convex function, then it has a global minimum that satisfies ∇y(θc ) = 0, i.e., at the minimum θ˙c = 0 and adaptation stops. The implementation of (1.2.9) requires the on-line generation of the sensitivity functions ∇y that usually depend on the unknown plant parameters and are, therefore, unavailable. In these cases, approximate values of the sensitivity functions are used instead of the actual ones. One type of approximation is to use some a priori knowledge about the plant parameters to compute the sensitivity functions. A popular method for computing the approximate sensitivity functions is the so-called MIT rule. With this rule the unknown parameters that are needed to generate the sensitivity functions are replaced by their on-line estimates. Unfortunately, with the use of approximate sensitivity functions, it is not possible, in general, to prove global closed-loop stability and convergence of the tracking error to zero. In simulations, however, it was observed that the MIT rule and other approximation techniques performed well when the adaptive gain γ and the magnitude of the reference input signal are small. Averaging techniques are used in [135] to confirm these observations and establish local stability for a certain class of reference input signals. Globally,

1.2. ADAPTIVE CONTROL

19

however, the schemes based on the MIT rule and other approximations may go unstable. Examples of instability are presented in [93, 187, 202]. We illustrate the use of the MIT rule for the design of an MRAC scheme for the plant y¨ = −a1 y˙ − a2 y + u (1.2.10) where a1 and a2 are the unknown plant parameters, and y˙ and y are available for measurement. The reference model to be matched by the closed loop plant is given by y¨m = −2y˙ m − ym + r

(1.2.11)

u = θ1∗ y˙ + θ2∗ y + r

(1.2.12)

θ1∗ = a1 − 2, θ2∗ = a2 − 1

(1.2.13)

The control law

where

will achieve perfect model following. The equation (1.2.13) is referred to as the matching equation. Because a1 and a2 are unknown, the desired values of the controller parameters θ1∗ and θ2∗ cannot be calculated from (1.2.13). Therefore, instead of (1.2.12) we use the control law u = θ1 y˙ + θ2 y + r

(1.2.14)

where θ1 and θ2 are adjusted using the MIT rule as ∂y ∂y , θ˙2 = −γe1 θ˙1 = −γe1 ∂θ1 ∂θ2

(1.2.15)

where e1 = y−ym . To implement (1.2.15), we need to generate the sensitivity ∂y ∂y functions ∂θ , on-line. 1 ∂θ2 Using (1.2.10) and (1.2.14) we obtain ∂ y¨ ∂ y˙ ∂y ∂ y˙ ∂y = −a1 − a2 + y˙ + θ1 + θ2 ∂θ1 ∂θ1 ∂θ1 ∂θ1 ∂θ1

(1.2.16)

∂ y¨ ∂ y˙ ∂y ∂ y˙ ∂y = −a1 − a2 + y + θ1 + θ2 ∂θ2 ∂θ2 ∂θ2 ∂θ2 ∂θ2

(1.2.17)

20

CHAPTER 1. INTRODUCTION

If we now assume that the rate of adaptation is slow, i.e., θ˙1 and θ˙2 are small, and the changes of y¨ and y˙ with respect to θ1 and θ2 are also small, we can interchange the order of differentiation to obtain d2 ∂y d ∂y ∂y = (θ1 − a1 ) + (θ2 − a2 ) + y˙ 2 dt ∂θ1 dt ∂θ1 ∂θ1

(1.2.18)

d2 ∂y d ∂y ∂y = (θ1 − a1 ) + (θ2 − a2 ) +y 2 dt ∂θ2 dt ∂θ2 ∂θ2

(1.2.19)

which we may rewrite as 1 ∂y = 2 y˙ ∂θ1 p − (θ1 − a1 )p − (θ2 − a2 )

(1.2.20)

∂y 1 = 2 y ∂θ2 p − (θ1 − a1 )p − (θ2 − a2 )

(1.2.21)

4

d where p(·) = dt (·) is the differential operator. Because a1 and a2 are unknown, the above sensitivity functions cannot be used. Using the MIT rule, we replace a1 and a2 with their estimates a ˆ1 and a ˆ2 in the matching equation (1.2.13), i.e., we relate the estimates a ˆ1 and a ˆ2 with θ1 and θ2 using

a ˆ1 = θ1 + 2, a ˆ2 = θ2 + 1

(1.2.22)

and obtain the approximate sensitivity functions ∂y 1 ∂y 1 ' 2 y, ˙ ' 2 y ∂θ1 p + 2p + 1 ∂θ2 p + 2p + 1

(1.2.23)

The equations given by (1.2.23) are known as the sensitivity filters or models, and can be easily implemented to generate the approximate sensitivity functions for the adaptive law (1.2.15). As shown in [93, 135], the MRAC scheme based on the MIT rule is locally stable provided the adaptive gain γ is small, the reference input signal has a small amplitude and sufficient number of frequencies, and the initial conditions θ1 (0) and θ2 (0) are close to θ1∗ and θ2∗ respectively. For larger γ and θ1 (0) and θ2 (0) away from θ1∗ and θ2∗ , the MIT rule may lead to instability and unbounded signal response.

1.2. ADAPTIVE CONTROL

21

The lack of stability of MIT rule based adaptive control schemes prompted several researchers to look for different methods of designing adaptive laws. These methods include the positivity and Lyapunov design approach, and the gradient and least-squares methods that are based on the minimization of certain estimation error criteria. These methods are studied in detail in Chapters 4 and 8, and are briefly described below. (ii) Positivity and Lyapunov design This method of developing adaptive laws is based on the direct method of Lyapunov and its relationship with positive real functions. In this approach, the problem of designing an adaptive law is formulated as a stability problem where the differential equation of the adaptive law is chosen so that certain stability conditions based on Lyapunov theory are satisfied. The adaptive law developed is very similar to that based on the sensitivity method. The only difference is that the sensitivity functions in the approach (i) are replaced with ones that can be generated on-line. In addition, the Lyapunov-based adaptive control schemes have none of the drawbacks of the MIT rule-based schemes. The design of adaptive laws using Lyapunov’s direct method was suggested by Grayson [76], Parks [187], and Shackcloth and Butchart [202] in the early 1960s. The method was subsequently advanced and generalized to a wider class of plants by Phillipson [188], Monopoli [149], Narendra [172], and others. A significant part of Chapters 4 and 8 will be devoted to developing adaptive laws using the Lyapunov design approach. (iii) Gradient and least-squares methods based on estimation error cost criteria The main drawback of the sensitivity methods used in the 1960s is that the minimization of the performance cost function led to sensitivity functions that are not implementable. One way to avoid this drawback is to choose a cost function criterion that leads to sensitivity functions that are available for measurement. A class of such cost criteria is based on an error referred to as the estimation error that provides a measure of the discrepancy between the estimated and actual parameters. The relationship of the estimation error with the estimated parameters is chosen so that the cost function is convex, and its gradient with respect to the estimated parameters is implementable.

22

CHAPTER 1. INTRODUCTION

Several different cost criteria may be used, and methods, such as the gradient and least-squares, may be adopted to generate the appropriate sensitivity functions. As an example, let us design the adaptive law for the direct MRAC law (1.2.14) for the plant (1.2.10). We first rewrite the plant equation in terms of the desired controller parameters given by (1.2.13), i.e., we substitute for a1 = 2 + θ1∗ , a2 = 1 + θ2∗ in (1.2.10) to obtain y¨ = −2y˙ − y − θ1∗ y˙ − θ2∗ y + u

(1.2.24)

which may be rewritten as y = θ1∗ y˙ f + θ2∗ yf + uf

(1.2.25)

where y˙ f = −

s2

1 1 1 y, ˙ yf = − 2 y, uf = 2 u + 2s + 1 s + 2s + 1 s + 2s + 1

(1.2.26)

are signals that can be generated by filtering. If we now replace θ1∗ and θ2∗ with their estimates θ1 and θ2 in equation (1.2.25), we will obtain, yˆ = θ1 y˙ f + θ2 yf + uf

(1.2.27)

where yˆ is the estimate of y based on the estimate θ1 and θ2 of θ1∗ and θ2∗ . The error 4 ε1 = y − yˆ = y − θ1 y˙ f − θ2 yf − uf (1.2.28) is, therefore, a measure of the discrepancy between θ1 , θ2 and θ1∗ , θ2∗ , respectively. We refer to it as the estimation error. The estimates θ1 and θ2 can now be adjusted in a direction that minimizes a certain cost criterion that involves ε1 . A simple such criterion is J(θ1 , θ2 ) =

ε21 1 = (y − θ1 y˙ f − θ2 yf − uf )2 2 2

(1.2.29)

which is to be minimized with respect to θ1 , θ2 . It is clear that J(θ1 , θ2 ) is a convex function of θ1 , θ2 and, therefore, the minimum is given by ∇J = 0.

1.3. A BRIEF HISTORY

23

If we now use the gradient method to minimize J(θ1 , θ2 ), we obtain the adaptive laws ∂J ∂J θ˙1 = −γ1 = γ1 ε1 y˙ f , θ˙2 = −γ2 = γ2 ε1 yf ∂θ1 ∂θ2

(1.2.30)

where γ1 , γ2 > 0 are the adaptive gains and ε1 , y˙ f , yf are all implementable signals. Instead of (1.2.29), one may use a different cost criterion for ε1 and a different minimization method leading to a wide class of adaptive laws. In Chapters 4 to 9 we will examine the stability properties of a wide class of adaptive control schemes that are based on the use of estimation error criteria, and gradient and least-squares type of optimization techniques.

1.3

A Brief History

Research in adaptive control has a long history of intense activities that involved debates about the precise definition of adaptive control, examples of instabilities, stability and robustness proofs, and applications. Starting in the early 1950s, the design of autopilots for high-performance aircraft motivated an intense research activity in adaptive control. Highperformance aircraft undergo drastic changes in their dynamics when they fly from one operating point to another that cannot be handled by constant-gain feedback control. A sophisticated controller, such as an adaptive controller, that could learn and accommodate changes in the aircraft dynamics was needed. Model reference adaptive control was suggested by Whitaker et al. in [184, 235] to solve the autopilot control problem. The sensitivity method and the MIT rule was used to design the adaptive laws of the various proposed adaptive control schemes. An adaptive pole placement scheme based on the optimal linear quadratic problem was suggested by Kalman in [96]. The work on adaptive flight control was characterized by “a lot of enthusiasm, bad hardware and non-existing theory” [11]. The lack of stability proofs and the lack of understanding of the properties of the proposed adaptive control schemes coupled with a disaster in a flight test [219] caused the interest in adaptive control to diminish.

24

CHAPTER 1. INTRODUCTION

The 1960s became the most important period for the development of control theory and adaptive control in particular. State space techniques and stability theory based on Lyapunov were introduced. Developments in dynamic programming [19, 20], dual control [53] and stochastic control in general, and in system identification and parameter estimation [13, 229] played a crucial role in the reformulation and redesign of adaptive control. By 1966 Parks and others found a way of redesigning the MIT rule-based adaptive laws used in the MRAC schemes of the 1950s by applying the Lyapunov design approach. Their work, even though applicable to a special class of LTI plants, set the stage for further rigorous stability proofs in adaptive control for more general classes of plant models. The advances in stability theory and the progress in control theory in the 1960s improved the understanding of adaptive control and contributed to a strong renewed interest in the field in the 1970s. On the other hand, the simultaneous development and progress in computers and electronics that made the implementation of complex controllers, such as the adaptive ones, feasible contributed to an increased interest in applications of adaptive control. The 1970s witnessed several breakthrough results in the design of adaptive control. MRAC schemes using the Lyapunov design approach were designed and analyzed in [48, 153, 174]. The concepts of positivity and hyperstability were used in [123] to develop a wide class of MRAC schemes with well-established stability properties. At the same time parallel efforts for discrete-time plants in a deterministic and stochastic environment produced several classes of adaptive control schemes with rigorous stability proofs [72, 73]. The excitement of the 1970s and the development of a wide class of adaptive control schemes with well established stability properties was accompanied by several successful applications [80, 176, 230]. The successes of the 1970s, however, were soon followed by controversies over the practicality of adaptive control. As early as 1979 it was pointed out that the adaptive schemes of the 1970s could easily go unstable in the presence of small disturbances [48]. The nonrobust behavior of adaptive control became very controversial in the early 1980s when more examples of instabilities were published demonstrating lack of robustness in the presence of unmodeled dynamics or bounded disturbances [85, 197]. This stimulated many researchers, whose objective was to understand the mechanisms of instabilities and find ways to counteract them. By the mid 1980s, several

1.3. A BRIEF HISTORY

25

new redesigns and modifications were proposed and analyzed, leading to a body of work known as robust adaptive control. An adaptive controller is defined to be robust if it guarantees signal boundedness in the presence of “reasonable” classes of unmodeled dynamics and bounded disturbances as well as performance error bounds that are of the order of the modeling error. The work on robust adaptive control continued throughout the 1980s and involved the understanding of the various robustness modifications and their unification under a more general framework [48, 87, 84]. The solution of the robustness problem in adaptive control led to the solution of the long-standing problem of controlling a linear plant whose parameters are unknown and changing with time. By the end of the 1980s several breakthrough results were published in the area of adaptive control for linear time-varying plants [226]. The focus of adaptive control research in the late 1980s to early 1990s was on performance properties and on extending the results of the 1980s to certain classes of nonlinear plants with unknown parameters. These efforts led to new classes of adaptive schemes, motivated from nonlinear system theory [98, 99] as well as to adaptive control schemes with improved transient and steady-state performance [39, 211]. Adaptive control has a rich literature full with different techniques for design, analysis, performance, and applications. Several survey papers [56, 183], and books and monographs [3, 15, 23, 29, 48, 55, 61, 73, 77, 80, 85, 94, 105, 123, 144, 169, 172, 201, 226, 229, 230] have already been published. Despite the vast literature on the subject, there is still a general feeling that adaptive control is a collection of unrelated technical tools and tricks. The purpose of this book is to unify the various approaches and explain them in a systematic and tutorial manner.

Chapter 2

Models for Dynamic Systems 2.1

Introduction

In this chapter, we give a brief account of various models and parameterizations of LTI systems. Emphasis is on those ideas that are useful in studying the parameter identification and adaptive control problems considered in subsequent chapters. We begin by giving a summary of some canonical state space models for LTI systems and of their characteristics. Next we study I/O descriptions for the same class of systems by using transfer functions and differential operators. We express transfer functions as ratios of two polynomials and present some of the basic properties of polynomials that are useful for control design and system modeling. systems that we express in a form in which parameters, such as coefficients of polynomials in the transfer function description, are separated from signals formed by filtering the system inputs and outputs. These parametric models and their properties are crucial in parameter identification and adaptive control problems to be studied in subsequent chapters. The intention of this chapter is not to give a complete picture of all aspects of LTI system modeling and representation, but rather to present a summary of those ideas that are used in subsequent chapters. For further discussion on the topic of modeling and properties of linear systems, we refer the reader to several standard books on the subject starting with the elementary ones [25, 41, 44, 57, 121, 180] and moving to the more advanced 26

2.2. STATE-SPACE MODELS

27

ones [30, 42, 95, 198, 237, 238].

2.2 2.2.1

State-Space Models General Description

Many systems are described by a set of differential equations of the form x(t) ˙ = f (x(t), u(t), t),

x(t0 ) = x0

y(t) = g(x(t), u(t), t)

(2.2.1)

where t is the time variable x(t) is an n-dimensional vector with real elements that denotes the state of the system u(t) is an r-dimensional vector with real elements that denotes the input variable or control input of the system y(t) is an l-dimensional vector with real elements that denotes the output variables that can be measured f, g are real vector valued functions n is the dimension of the state x called the order of the system x(t0 ) denotes the value of x(t) at the initial time t = t0 ≥ 0 When f, g are linear functions of x, u, (2.2.1) takes the form x˙ = A(t)x + B(t)u,

x(t0 ) = x0

y = C > (t)x + D(t)u

(2.2.2)

where A(t) ∈ Rn×n , B(t) ∈ Rn×r , C(t) ∈ Rn×l , and D(t) ∈ Rl×r are matrices with time-varying elements. If in addition to being linear, f, g do not depend on time t, we have x˙ = Ax + Bu, >

y = C x + Du

x(t0 ) = x0 (2.2.3)

where A, B, C, and D are matrices of the same dimension as in (2.2.2) but with constant elements.

28

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS

We refer to (2.2.2) as the finite-dimensional linear time-varying (LTV) system and to (2.2.3) as the finite dimensional LTI system. The solution x(t), y(t) of (2.2.2) is given by x(t) = Φ(t, t0 )x(t0 ) +

Z t t0

Φ(t, τ )B(τ )u(τ )dτ

y(t) = C > (t)x(t) + D(t)u(t)

(2.2.4)

where Φ(t, t0 ) is the state transition matrix defined as a matrix that satisfies the linear homogeneous matrix equation ∂Φ(t, t0 ) = A(t)Φ(t, t0 ), ∂t

Φ(t0 , t0 ) = I

For the LTI system (2.2.3), Φ(t, t0 ) depends only on the difference t − t0 , i.e., Φ(t, t0 ) = Φ(t − t0 ) = eA(t−t0 ) and the solution x(t), y(t) of (2.2.3) is given by x(t) = eA(t−t0 ) x0 +

Z t t0

eA(t−τ ) Bu(τ )dτ

y(t) = C > x(t) + Du(t)

(2.2.5)

where eAt can be identified to be eAt = L−1 [(sI − A)−1 ] where L−1 denotes the inverse Laplace transform and s is the Laplace variable. Usually the matrix D in (2.2.2), (2.2.3) is zero, because in most physical systems there is no direct path of nonzero gain between the inputs and outputs. In this book, we are concerned mainly with LTI, SISO systems with D = 0. In some chapters and sections, we will also briefly discuss systems of the form (2.2.2) and (2.2.3).

2.2. STATE-SPACE MODELS

2.2.2

29

Canonical State-Space Forms

Let us consider the SISO, LTI system x˙ = Ax + Bu,

x(t0 ) = x0

y = C >x

(2.2.6)

where x ∈ Rn . The controllability matrix Pc of (2.2.6) is defined by 4

Pc = [B, AB, . . . , An−1 B] A necessary and sufficient condition for the system (2.2.6) to be completely controllable is that Pc is nonsingular. If (2.2.6) is completely controllable, the linear transformation xc = Pc−1 x

(2.2.7)

transforms (2.2.6) into the controllability canonical form 

x˙ c

   =    

y =

0 0 ··· 1 0 ··· 0 1 ··· .. .. . . 0 0 ···

0 0 0

−a0 −a1 −a2 .. .





       xc +       

1 −an−1

1 0 0 .. .

    u   

(2.2.8)

0

Cc> xc

where the ai ’s are the coefficients of the characteristic equation of A, i.e., det(sI − A) = sn + an−1 sn−1 + · · · + a0 and Cc> = C > Pc . If instead of (2.2.7), we use the transformation xc = M −1 Pc−1 x where

    M =   

1 an−1 0 1 .. .. . . 0 0 0 0

· · · a2 a1 · · · a3 a2 . .. .. . .. . · · · 1 an−1 ··· 0 1

(2.2.9)        

30

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS

we obtain the following controller canonical form 

x˙ c

   =    

−an−1 −an−2 1 0 0 1 .. .. . . 0 0





· · · −a1 −a0  ··· 0 0      ··· 0 0  xc +    ..   .. .  .  ··· 1 0

1 0 0 .. .

     u (2.2.10)   

0

y = C0> xc where C0> = C > Pc M . By rearranging the elements of the state vector xc , (2.2.10) may be written in the following form that often appears in books on linear system theory 

x˙ c

   =    

0 0 .. .

1 0 .. .

0 1

··· ··· .. .

0 0

0 0 ··· 1 −a0 −a1 · · · −an−2 −an−1





0 0 .. .

       xc +        0

     u (2.2.11)   

1

y = C1> xc where C1 is defined appropriately. The observability matrix Po of (2.2.6) is defined by  4

 

Po =   

C> C >A .. .

     

(2.2.12)

C > An−1 A necessary and sufficient condition for the system (2.2.6) to be completely observable is that Po is nonsingular. By following the dual of the arguments presented earlier for the controllability and controller canonical forms, we arrive at observability and observer forms provided Po is nonsingular [95], i.e., the observability canonical form of (2.2.6) obtained by using the trans-

2.2. STATE-SPACE MODELS

31

formation xo = Po x is        

x˙ o =

0 0 .. .

1 0 .. .

0 1

··· ··· .. .

0 0 .. .

0 0 ··· 1 −a0 −a1 · · · −an−2 −an−1

     xo + Bo u   

(2.2.13)

y = [1, 0, . . . , 0]xo and the observer canonical form is 

x˙ o

   =    

−an−1 1 0 · · · −an−2 0 1 · · · .. .. .. . . . −a1 0 0 · · · −a0 0 0 · · ·



0 0  

  xo + B1 u   1 

(2.2.14)

0

y = [1, 0, . . . , 0]xo where Bo , B1 may be different. If the rank of the controllability matrix Pc for the nth-order system (2.2.6) is less than n, then (2.2.6) is said to be uncontrollable. Similarly, if the rank of the observability matrix Po is less than n, then (2.2.6) is unobservable. The system represented by (2.2.8) or (2.2.10) or (2.2.11) is completely controllable but not necessarily observable. Similarly, the system represented by (2.2.13) or (2.2.14) is completely observable but not necessarily controllable. If the nth-order system (2.2.6) is either unobservable or uncontrollable then its I/O properties for zero initial state, i.e., x0 = 0 are completely characterized by a lower order completely controllable and observable system x˙ co = Aco xco + Bco u, y =

> Cco xco

xco (t0 ) = 0 (2.2.15)

where xco ∈ Rnr and nr < n. It turns out that no further reduction in the order of (2.2.15) is possible without affecting the I/O properties for all inputs.

32

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS m1 θ1

m2 θ2

l1 u

l2

M

Figure 2.1 Cart with two inverted pendulums

For this reason (2.2.15) is referred to as the minimal state-space representation of the system to be distinguished from the nonminimal state-space representation that corresponds to either an uncontrollable or unobservable system. A minimal state space model does not describe the uncontrollable or unobservable parts of the system. These parts may lead to some unbounded states in the nonminimal state-space representation of the system if any initial condition associated with these parts is nonzero. If, however, the uncontrollable or unobservable parts are asymptotically stable [95], they will decay to zero exponentially fast, and their effect may be ignored in most applications. A system whose uncontrollable parts are asymptotically stable is referred to as stabilizable, and the system whose unobservable parts are asymptotically stable is referred to as detectable [95]. Example 2.2.1 Let us consider the cart with the two inverted pendulums shown in Figure 2.1, where M is the mass of the cart, m1 and m2 are the masses of the bobs, and l1 and l2 are the lengths of the pendulums, respectively. Using Newton’s law and assuming small angular deviations of |θ1 |, |θ2 |, the equations of motions are given by M v˙ = −m1 gθ1 − m2 gθ2 + u m1 (v˙ + l1 θ¨1 ) = m1 gθ1 m2 (v˙ + l2 θ¨2 ) = m2 gθ2 where v is the velocity of the cart, u is an external force, and g is the acceleration due to gravity. To simplify the algebra, let us assume that m1 = m2 = 1kg and M = 10m1 . If we now let x1 = θ1 , x2 = θ˙1 , x3 = θ1 − θ2 , x4 = θ˙1 − θ˙2 be the state variables, we obtain the following state-space representation for the system: x˙ = Ax + Bu

2.2. STATE-SPACE MODELS where x = [x1 , x2 , x3 , x4 ]>  0  1.2α 1 A=  0 1.2(α1 − α2 )

1 0 0 0

33

 0 0 −0.1α1 0  , 0 1  α2 − 0.1(α1 − α2 ) 0



 0   β1  B=   0 β1 − β2

0.1 and α1 = lg1 , α2 = lg2 , β1 = − 0.1 l1 , and β2 = − l2 . The controllability matrix of the system is given by

Pc = [B, AB, A2 B, A3 B] We can verify that detPc =

(0.011)2 g 2 (l1 − l2 )2 l14 l24

which implies that the system is controllable if and only if l1 6= l2 . Let us now assume that θ1 is the only variable that we measure, i.e., the measured output of the system is y = C >x where C = [1, 0, 0, 0]> . The observability matrix of the system based on this output is given by   C>  C >A   Po =   C > A2  C > A3 By performing the calculations, we verify that detPo = 0.01

g2 l12

which implies that the system is always observable from y = θ1 . When l1 = l2 , the system is uncontrollable. In this case, α1 = α2 , β1 = β2 , and the matrix A and vector B become     0 1 0 0 0  1.2α1 0 −0.1α1 0     , B =  β1  A=  0   0 0 1 0  0 0 α1 0 0 indicating that the control input u cannot influence the states x3 , x4 . It can be verified that for x3 (0), x4 (0) 6= 0, all the states will grow to infinity for all possible inputs u. For l1 = l2 , the control of the two identical pendulums is possible provided the initial angles and angular velocities are identical, i.e., θ1 (0) = θ2 (0) and θ˙1 (0) = θ˙2 (0), which imply that x3 (0) = x4 (0) = 0. 5

34

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS

2.3 2.3.1

Input/Output Models Transfer Functions

Transfer functions play an important role in the characterization of the I/O properties of LTI systems and are widely used in classical control theory. We define the transfer function of an LTI system by starting with the differential equation that describes the dynamic system. Consider a system described by the nth-order differential equation y (n) (t)+an−1 y (n−1) (t)+· · ·+a0 y(t) = bm u(m) (t)+bm−1 u(m−1) (t)+· · ·+b0 u(t) (2.3.1) 4

4

i

i

d (i) (t) = d u(t); u(t) is the input variable, where y (i) (t) = dt i y(t), and u dti and y(t) is the output variable; the coefficients ai , bj , i = 0, 1 . . . , n − 1, j = 0, 1, . . . , m are constants, and n and m are constant integers. To obtain the transfer function of the system (2.3.1), we take the Laplace transform on both sides of the equation and assume zero initial conditions, i.e.,

(sn + an−1 sn−1 + · · · + a0 )Y (s) = (bm sm + bm−1 sm−1 + · · · + b0 )U (s) where s is the Laplace variable. The transfer function G(s) of (2.3.1) is defined as bm sm + bm−1 sm−1 + · · · + b0 4 Y (s) G(s) = = (2.3.2) U (s) sn + an−1 sn−1 + · · · + a0 The inverse Laplace g(t) of G(s), i.e., 4

g(t) = L−1 [G(s)] is known as the impulse response of the system (2.3.1) and y(t) = g(t) ∗ u(t) where ∗ denotes convolution. When u(t) = δ∆ (t) where δ∆ (t) is the delta function defined as I(t) − I(t − ²) δ∆ (t) = lim ²→0 ² where I(t) is the unit step function, then y(t) = g(t) ∗ δ∆ (t) = g(t)

2.3. INPUT/OUTPUT MODELS

35

Therefore, when the input to the LTI system is a delta function (often referred to as a unit impulse) at t = 0, the output of the system is equal to g(t), the impulse response. We say that G(s) is proper if G(∞) is finite i.e., n ≥ m; strictly proper if G(∞) = 0 , i.e., n > m; and biproper if n = m. The relative degree n∗ of G(s) is defined as n∗ = n − m, i.e., n∗ = degree of denominator - degree of numerator of G(s). The characteristic equation of the system (2.3.1) is defined as the equation sn + an−1 sn−1 + · · · + a0 = 0. In a similar way, the transfer function may be defined for the LTI system in the state space form (2.2.3), i.e., taking the Laplace transform on each side of (2.2.3) we obtain sX(s) − x(0) = AX(s) + BU (s) Y (s) = C > X(s) + DU (s) or

³

(2.3.3)

´

Y (s) = C > (sI − A)−1 B + D U (s) + C > (sI − A)−1 x(0) Setting the initial conditions to zero, i.e., x(0) = 0 we get Y (s) = G(s)U (s)

(2.3.4)

where G(s) = C > (sI − A)−1 B + D is referred to as the transfer function matrix in the case of multiple inputs and outputs and simply as the transfer function in the case of SISO systems. We may also represent G(s) as G(s) =

C > {adj(sI − A)}B +D det(sI − A)

(2.3.5)

where adjQ denotes the adjoint of the square matrix Q ∈ Rn×n . The (i, j) element qij of adjQ is given by qij = (−1)i+j det(Qji );

i, j = 1, 2, . . . n

36

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS

where Qji ∈ R(n−1)×(n−1) is a submatrix of Q obtained by eliminating the jth row and the ith column of the matrix Q. It is obvious from (2.3.5) that the poles of G(s) are included in the eigenvalues of A. We say that A is stable if all its eigenvalues lie in Re[s] < 0 in which case G(s) is a stable transfer function. It follows that det(sI−A) = 0 is the characteristic equation of the system with transfer function given by (2.3.5). In (2.3.3) and (2.3.4) we went from a state-space representation to a transfer function description in a straightforward manner. The other way, i.e., from a proper transfer function description to a state-space representation, is not as straightforward. It is true, however, that for every proper transfer function G(s) there exists matrices A, B, C, and D such that G(s) = C > (sI − A)−1 B + D As an example, consider a system with the transfer function G(s) =

Y (s) bm sm + bm−1 sm−1 + · · · + b0 = n n−1 s + an−1 s + · · · + a0 U (s)

where n > m. Then the system may be represented in the controller form     x˙ =    

−an−1 −an−2 1 0 0 1 .. .. . . 0 0





· · · −a1 −a0  ··· 0 0      ··· 0 0 x +    ..   .. .   . ··· 1 0

1 0 .. .



   u   0 

(2.3.6)

0

y = [0, 0, . . . , bm , . . . , b1 , b0 ]x or in the observer form     x˙ =    

−an−1 1 0 · · · −an−2 0 1 · · · .. .. .. . . . −a1 0 0 · · · −a0 0 0 · · ·

y = [1, 0, . . . , 0]x

0 0 .. .





    x +       1  

0



0  .   ..  bm .. . b0

  u   

(2.3.7)

2.3. INPUT/OUTPUT MODELS

37

One can go on and generate many different state-space representations describing the I/O properties of the same system. The canonical forms in (2.3.6) and (2.3.7), however, have some important properties that we will use in later chapters. For example, if we denote by (Ac , Bc , Cc ) and (Ao , Bo , Co ) the corresponding matrices in the controller form (2.3.6) and observer form (2.3.7), respectively, we establish the relations 4

[adj(sI − Ac )]Bc = [sn−1 , . . . , s, 1]> = αn−1 (s)

(2.3.8)

> Co> adj(sI − Ao ) = [sn−1 , . . . , s, 1] = αn−1 (s)

(2.3.9)

whose right-hand sides are independent of the coefficients of G(s). Another important property is that in the triples (Ac , Bc , Cc ) and (Ao , Bo , Co ), the n + m + 1 coefficients of G(s) appear explicitly, i.e., (Ac , Bc , Cc ) (respectively (Ao , Bo , Co )) is completely characterized by n + m + 1 parameters, which are equal to the corresponding coefficients of G(s). If G(s) has no zero-pole cancellations then both (2.3.6) and (2.3.7) are minimal state-space representations of the same system. If G(s) has zeropole cancellations, then (2.3.6) is unobservable, and (2.3.7) is uncontrollable. If the zero-pole cancellations of G(s) occur in Re[s] < 0, i.e., stable poles are cancelled by stable zeros, then (2.3.6) is detectable, and (2.3.7) is stabilizable. Similarly, a system described by a state-space representation is unobservable or uncontrollable, if and only if the transfer function of the system has zeropole cancellations. If the unobservable or uncontrollable parts of the system are asymptotically stable, then the zero-pole cancellations occur in Re[s] < 0. An alternative approach for representing the differential equation (2.3.1) is by using the differential operator 4

p(·) =

d(·) dt

which has the following properties: (i) p(x) = x; ˙

(ii) p(xy) = xy ˙ + xy˙ 4 dx(t) dt . 1 p is defined

where x and y are any differentiable functions of time and x˙ = The inverse of the operator p denoted by p−1 or simply by as Z t 1 4 (x) = x(τ )dτ + x(0) ∀t ≥ 0 p 0

38

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS

where x(t) is an integrable function of time. The operators p, p1 are related to the Laplace operator s by the following equations L {p(x)}|x(0)=0 = sX(s) 1 1 L{ (x)} |x(0)=0 = X(s) p s where L is the Laplace transform and x(t) is any differentiable function of time. Using the definition of the differential operator, (2.3.1) may be written in the compact form R(p)(y) = Z(p)(u) (2.3.10) where R(p) = pn + an−1 pn−1 + · · · + a0 Z(p) = bm pm + bm−1 pm−1 + · · · + b0 are referred to as the polynomial differential operators [226]. Equation (2.3.10) has the same form as R(s)Y (s) = Z(s)U (s)

(2.3.11)

obtained by taking the Laplace transform on both sides of (2.3.1) and assuming zero initial conditions. Therefore, for zero initial conditions one can go from representation (2.3.10) to (2.3.11) and vice versa by simply replacing s with p or p with s appropriately. For example, the system Y (s) =

s + b0 U (s) s2 + a0

may be written as (p2 + a0 )(y) = (p + b0 )(u) with y(0) = y(0) ˙ = 0, u(0) = 0 or by abusing notation (because we never defined the operator (p2 + a0 )−1 ) as y(t) =

p + b0 u(t) p2 + a0

Because of the similarities of the forms of (2.3.11) and (2.3.10), we will use s to denote both the differential operator and Laplace variable and express the system (2.3.1) with zero initial conditions as y=

Z(s) u R(s)

(2.3.12)

2.3. INPUT/OUTPUT MODELS

39

where y and u denote Y (s) and U (s), respectively, when s is taken to be the Laplace operator, and y and u denote y(t) and u(t), respectively, when s is taken to be the differential operator. Z(s) We will often refer to G(s) = R(s) in (2.3.12) as the filter with input u(t) and output y(t). Example 2.3.1 Consider the system of equations describing the motion of the cart with the two pendulums given in Example 2.2.1, where y = θ1 is the only measured output. Eliminating the variables θ1 , θ2 , and θ˙2 by substitution, we obtain the fourth order differential equation y (4) − 1.1(α1 + α2 )y (2) + 1.2α1 α2 y = β1 u(2) − α1 β2 u where αi , βi , i = 1, 2 are as defined in Example 2.2.1, which relates the input u with the measured output y. Taking the Laplace transform on each side of the equation and assuming zero initial conditions, we obtain [s4 − 1.1(α1 + α2 )s2 + 1.2α1 α2 ]Y (s) = (β1 s2 − α1 β2 )U (s) Therefore, the transfer function of the system from u to y is given by Y (s) β1 s2 − α1 β2 = 4 = G(s) U (s) s − 1.1(α1 + α2 )s2 + 1.2α1 α2 For l1 = l2 , we have α1 = α2 , β1 = β2 , and G(s) =

s4

β1 (s2 − α1 ) β1 (s2 − α1 ) = 2 2 2 − 2.2α1 s + 1.2α1 (s − α1 )(s2 − 1.2α1 )

has two zero-pole cancellations. Because α1 > 0, one of the zero-pole cancellations occurs in Re[s] > 0 which indicates that any fourth-order state representation of the system with the above transfer function is not stabilizable. 5

2.3.2

Coprime Polynomials

The I/O properties of most of the systems studied in this book are represented by proper transfer functions expressed as the ratio of two polynomials in s with real coefficients, i.e., G(s) =

Z(s) R(s)

(2.3.13)

40

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS

where Z(s) = bm sm + bm−1 sm−1 + · · · + b0 , R(s) = sn + an−1 sn−1 + · · · + a0 and n ≥ m. The properties of the system associated with G(s) depend very much on the properties of Z(s) and R(s). In this section, we review some of the general properties of polynomials that are used for analysis and control design in subsequent chapters. Definition 2.3.1 Consider the polynomial X(s) = αn sn + αn−1 sn−1 + · · · + α0 . We say that X(s) is monic if αn = 1 and X(s) is Hurwitz if all the roots of X(s) = 0 are located in Re[s] < 0. We say that the degree of X(s) is n if the coefficient αn of sn satisfies αn 6= 0. Definition 2.3.2 A system with a transfer function given by (2.3.13) is referred to as minimum phase if Z(s) is Hurwitz; it is referred to as stable if R(s) is Hurwitz. As we mentioned in Section 2.3.1, a system representation is minimal if the corresponding transfer function has no zero-pole cancellations, i.e., if the numerator and denominator polynomials of the transfer function have no common factors other than a constant. The following definition is widely used in control theory to characterize polynomials with no common factors. Definition 2.3.3 Two polynomials a(s) and b(s) are said to be coprime (or relatively prime) if they have no common factors other than a constant. An important characterization of coprimeness of two polynomials is given by the following Lemma. Lemma 2.3.1 (Bezout Identity) Two polynomials a(s) and b(s) are coprime if and only if there exist polynomials c(s) and d(s) such that c(s)a(s) + d(s)b(s) = 1 For a proof of Lemma 2.3.1, see [73, 237]. The Bezout identity may have infinite number of solutions c(s) and d(s) for a given pair of coprime polynomials a(s) and b(s) as illustrated by the following example.

2.3. INPUT/OUTPUT MODELS

41

Example 2.3.2 Consider the coprime polynomials a(s) = s + 1, b(s) = s + 2. Then the Bezout identity is satisfied for c(s) = sn + 2sn−1 − 1, d(s) = −sn − sn−1 + 1 and any n ≥ 1.

5

Coprimeness is an important property that is often exploited in control theory for the design of control schemes for LTI systems. An important theorem that is very often used for control design and analysis is the following. Theorem 2.3.1 If a(s) and b(s) are coprime and of degree na and nb , respectively, where na > nb , then for any given arbitrary polynomial a∗ (s) of degree na∗ ≥ na , the polynomial equation a(s)l(s) + b(s)p(s) = a∗ (s)

(2.3.14)

has a unique solution l(s) and p(s) whose degrees nl and np , respectively, satisfy the constraints np < na , nl ≤ max(na∗ − na , nb − 1).

Proof From Lemma 2.3.1, there exist polynomials c(s) and d(s) such that a(s)c(s) + b(s)d(s) = 1

(2.3.15)

Multiplying Equation (2.3.15) on both sides by the polynomial a∗ (s), we obtain a∗ (s)a(s)c(s) + a∗ (s)b(s)d(s) = a∗ (s)

(2.3.16)

Let us divide a∗ (s)d(s) by a(s), i.e., a∗ (s)d(s) p(s) = r(s) + a(s) a(s) where r(s) is the quotient of degree na∗ + nd − na ; na∗ , na , and nd are the degrees of a∗ (s), a(s), and d(s), respectively, and p(s) is the remainder of degree np < na . We now use a∗ (s)d(s) = r(s)a(s) + p(s) to express the right-hand side of (2.3.16) as a∗ (s)a(s)c(s) + r(s)a(s)b(s) + p(s)b(s) = [a∗ (s)c(s) + r(s)b(s)]a(s) + p(s)b(s)

42

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS

and rewrite (2.3.16) as l(s)a(s) + p(s)b(s) = a∗ (s)

(2.3.17)

where l(s) = a∗ (s)c(s) + r(s)b(s). The above equation implies that the degree of l(s)a(s) = degree of (a∗ (s) − p(s)b(s)) ≤ max{na∗ , np + nb }. Hence, the degree of l(s), denoted by nl , satisfies nl ≤ max{na∗ − na , np + nb − na }. We, therefore, established that polynomials l(s) and p(s) of degree nl ≤ max{na∗ − na , np + nb − na } and np < na respectively exist that satisfy (2.3.17). Because np < na implies that np ≤ na − 1, the degree nl also satisfies nl ≤ max{n∗a − na , nb − 1}. We show the uniqueness of l(s) and p(s) by proceeding as follows: We suppose that (l1 (s), p1 (s)), (l2 (s), p2 (s)) are two solutions of (2.3.17) that satisfy the degree constraints np < na , nl ≤ max{na∗ − na , nb − 1}, i.e., a(s)l1 (s) + b(s)p1 (s) = a∗ (s),

a(s)l2 (s) + b(s)p2 (s) = a∗ (s)

Subtracting one equation from another, we have a(s)(l1 (s) − l2 (s)) + b(s)(p1 (s) − p2 (s)) = 0

(2.3.18)

b(s) l2 (s) − l1 (s) = a(s) p1 (s) − p2 (s)

(2.3.19)

which implies that

Because np < na , equation (2.3.19) implies that b(s), a(s) have common factors that contradicts with the assumption that a(s) and b(s) are coprime. Thus, l1 (s) = l2 (s) and p1 (s) = p2 (s), which implies that the solution l(s) and p(s) of (2.3.17) is unique, and the proof is complete. 2

If no constraints are imposed on the degrees of l(s) and p(s), (2.3.14) has an infinite number of solutions. Equations of the form (2.3.14) are referred to as Diophantine equations and are widely used in the algebraic design of controllers for LTI plants. The following example illustrates the use of Theorem 2.3.1 for designing a stable control system. Example 2.3.3 Let us consider the following plant y=

s−1 u s3

(2.3.20)

We would like to choose the input u(t) so that the closed-loop characteristic equation of the plant is given by a∗ (s) = (s+1)5 , i.e., u is to be chosen so that the closed-loop plant is described by (s + 1)5 y = 0 (2.3.21)

2.3. INPUT/OUTPUT MODELS

43

Let us consider the control input in the form of u=−

p(s) y l(s)

(2.3.22)

where l(s) and p(s) are polynomials with real coefficients whose degrees and coefficients are to be determined. Using (2.3.22) in (2.3.20), we have the closed-loop plant s3 l(s)y = −(s − 1)p(s)y i.e., [l(s)s3 + p(s)(s − 1)]y = 0 If we now choose l(s) and p(s) to satisfy the Diophantine equation l(s)s3 + p(s)(s − 1) = (s + 1)5

(2.3.23)

then the closed-loop plant becomes the same as the desired one given by (2.3.21). Because (2.3.23) may have an infinite number of solutions for l(s), p(s), we use Theorem 2.3.1 to choose l(s) and p(s) with the lowest degree. According to Theorem 2.3.1, Equation (2.3.23) has a unique solution l(s), p(s) of degree equal to at most 2. Therefore, we assume that l(s), p(s) have the form l(s) = l2 s2 + l1 s + l0 p(s) = p2 s2 + p1 s + p0 which we use in (2.3.23) to obtain the following polynomial equation l2 s5 + l1 s4 + (l0 + p2 )s3 +(p1 − p2 )s2 + (p0 − p1 )s − p0 = s5 + 5s4 + 10s3 + 10s2 +5s +1 Equating the coefficients of the same powers of s on each side of the above equation, we obtain the algebraic equations  l2 = 1     l  1 =5   l0 + p2 = 10 p1 − p2 = 10     p  0 − p1 = 5   −p0 = 1 which have the unique solution of l2 = 1, l1 = 5, l0 = 26, p2 = −16, p1 = −6, p0 = −1. Hence, l(s) = s2 + 5s + 26, p(s) = −16s2 − 6s − 1 and from (2.3.22) the control input is given by

44

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS u=

16s2 + 6s + 1 y s2 + 5s + 26

5

Another characterization of coprimeness that we use in subsequent chapters is given by the following theorem: Theorem 2.3.2 (Sylvester’s Theorem) Two polynomials a(s) = an sn + an−1 sn−1 + · · · + a0 , b(s) = bn sn + bn−1 sn−1 + · · · + b0 are coprime if and only if their Sylvester matrix Se is nonsingular, where Se is defined to be the following 2n × 2n matrix: 

an  a  n−1

          4 Se=            

0 an

0 0

·

an−1

an

· · a1 a0 0 0 .. . .. . 0

· · · a1 a0 0

· · · · · · .. .

0

···

··· ..

.

· · · · · · · ..

.

0

0 0

0 bn 0 0 bn−1 bn .. . · bn−1 .. .. . . · · · 0 · · · an b1 · · an−1 b0 b1 · · 0 b0 · · 0 0 .. · · . 0 .. a0 a1 . 0 a0 0 0

0 0 bn · · · · · · .. . ···

··· ..

.

· · · · · · · .. 0

.

0 0

0 0 .. . .. . 0 bn



       ..  .   ·   ·   · bn−1  · ·   · ·    · ·     b0 b1 

0

b0 (2.3.24)

Proof If Consider the following polynomial equation a(s)c(s) + b(s)d(s) = 1

(2.3.25)

where c(s) = cn−1 sn−1 + cn−2 sn−2 + · · · + c0 , d(s) = dn−1 sn−1 + dn−2 sn−2 + · · · + d0 are some polynomials. Equating the coefficients of equal powers of s on both sides of (2.3.25), we obtain the algebraic equation Se p = e2n where e2n = [0, 0, . . . , 0, 1]> and p = [cn−1 , cn−2 , . . . , c0 , dn−1 , dn−2 , . . . , d0 ]>

(2.3.26)

2.3. INPUT/OUTPUT MODELS

45

Equations (2.3.25) and (2.3.26) are equivalent in the sense that any solution of (2.3.26) satisfies (2.3.25) and vice versa. Because Se is nonsingular, equation (2.3.26) has a unique solution for p. It follows that (2.3.25) also has a unique solution for c(s) and d(s) which according to Lemma 2.3.1 implies that a(s), b(s) are coprime. Only if We claim that if a(s) and b(s) are coprime, then for all nonzero polynomials p(s) and q(s) of degree np < n and nq < n, respectively, we have a(s)p(s) + b(s)q(s) 6≡ 0

(2.3.27)

If the claim is not true, there exists nonzero polynomials p1 (s) and q1 (s) of degree np1 < n and nq1 < n, respectively, such that a(s)p1 (s) + b(s)q1 (s) ≡ 0

(2.3.28)

Equation (2.3.28) implies that b(s)/a(s) can be expressed as b(s) p1 (s) =− a(s) q1 (s) which, because np1 < n and nq1 < n, implies that a(s) and b(s) have common factors, thereby contradicting the assumption that a(s), b(s) are coprime. Hence, our claim is true and (2.3.27) holds. Now (2.3.27) may be written as Se x 6= 0

(2.3.29)

where x ∈ R2n contains the coefficients of p(s), q(s). Because (2.3.27) holds for all nonzero p(s) and q(s) of degree np < n and nq < n, respectively, then (2.3.29) holds for all vectors x ∈ R2n with x 6= 0, which implies that Se is nonsingular. 2

The determinant of Se is known as the Sylvester resultant and may be used to examine the coprimeness of a given pair of polynomials. If the polynomials a(s) and b(s) in Theorem 2.3.2 have different degrees—say nb < na —then b(s) is expressed as a polynomial of degree na by augmenting it with the additional powers in s whose coefficients are taken to be equal to zero. Example 2.3.4 Consider the polynomials a(s) = s2 + 2s + 1, b(s) = s − 1 = 0s2 + s − 1

46

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS

Their Sylvester matrix is given by 

1  2 Se =   1 0

0 1 2 1

0 1 −1 0

 0 0   1  −1

Because detSe = 4 6= 0, a(s) and b(s) are coprime polynomials.

5

The properties of the Sylvester matrix are useful in solving a class of Diophantine equations of the form l(s)a(s) + p(s)b(s) = a∗ (s) for l(s) and p(s) where a(s), b(s), and a∗ (s) are given polynomials. For example, equation a(s)l(s) + b(s)p(s) = a∗ (s) with na = n, na∗ = 2n − 1, and nb = m < n implies the algebraic equation Se x = f

(2.3.30)

where Se ∈ R2n×2n is the Sylvester matrix of a(s), b(s), and x ∈ R2n is a vector with the coefficients of the polynomials l(s) and p(s) whose degree according to Theorem 2.3.1 is at most n − 1 and f ∈ R2n contains the coefficients of a∗ (s). Therefore, given a∗ (s), a(s), and b(s), one can solve (2.3.30) for x, the coefficient vector of l(s) and p(s). If a(s), b(s) are coprime, Se−1 exists and, therefore, the solution of (2.3.30) is unique and is given by x = Se−1 f If a(s), b(s) are not coprime, then Se is not invertible, and (2.3.30) has a solution if and only if the vector f is in the range of Se . One can show through algebraic manipulations that this condition is equivalent to the condition that a∗ (s) contains the common factors of a(s) and b(s). Example 2.3.5 Consider the same control design problem as in Example 2.3.3, s−1 where the control input u = − p(s) l(s) y is used to force the plant y = s3 u to satisfy (s + 1)5 y = 0. We have shown that the polynomials l(s) and p(s) satisfy the Diophantine equation l(s)a(s) + p(s)b(s) = (s + 1)5 (2.3.31)

2.4. PLANT PARAMETRIC MODELS where a(s) = s3 and b(s) = s − 1. and b(s) is  1  0   0 Se =   0   0 0

47

The corresponding Sylvester matrix Se of a(s) 0 1 0 0 0 0

0 0 1 0 0 0

0 0 1 −1 0 0

0 0 0 1 −1 0

0 0 0 0 1 −1

       

Because det Se = −1, we verify that a(s), b(s) are coprime. As in Example 2.3.3, we like to solve (2.3.31) for the unknown coefficients li , pi , i = 0, 1, 2 of the polynomials l(s) = l2 s2 + l1 s + l0 and p(s) = p2 s2 + p1 s + p0 . By equating the coefficients of equal powers of s on each side of (2.3.31), we obtain the algebraic equation Se x = f (2.3.32) where f = [1, 5, 10, 10, 5, 1]> and x = [l2 , l1 , l0 , p2 , p1 , p0 ]> . Because Se is a nonsingular matrix, the solution of (2.3.32) is given by >

x = Se−1 f = [1, 5, 26, −16, −6, −1, ]

which is the same as the solution we obtained in Example 2.3.3 (verify!).

2.4

5

Plant Parametric Models

Let us consider the plant represented by the following minimal state-space form: x˙ = Ax + Bu, x(0) = x0 (2.4.1) y = C >x where x ∈ Rn , u ∈ R1 , and y ∈ R1 and A, B, and C have the appropriate dimensions. The triple (A, B, C) consists of n2 +2n elements that are referred to as the plant parameters. If (2.4.1) is in one of the canonical forms studied in Section 2.2.2, then n2 elements of (A, B, C) are fixed to be 0 or 1 and at most 2n elements are required to specify the properties of the plant. These 2n elements are the coefficients of the numerator and denominator of the (s) transfer function YU (s) . For example, using the Laplace transform in (2.4.1), we obtain Y (s) = C > (sI − A)−1 BU (s) + C > (sI − A)−1 x0

48

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS

which implies that Y (s) =

Z(s) C > {adj(sI − A)} U (s) + x0 R(s) R(s)

(2.4.2)

where R(s) is a polynomial of degree n and Z(s) of degree at most n − 1. Setting x0 = 0, we obtain the transfer function description y=

Z(s) u R(s)

(2.4.3)

where without loss of generality, we can assume Z(s) and R(s) to be of the form Z(s) = bn−1 sn−1 + bn−2 sn−2 + · · · + b1 s + b0 n

n−1

R(s) = s + an−1 s

(2.4.4)

+ · · · + a1 s + a0

If Z(s) is of degree m < n−1, then the coefficients bi , i = n−1, n−2, . . . , m+1 are equal to zero. Equations (2.4.3) and (2.4.4) indicate that at most 2n parameters are required to uniquely specify the I/O properties of (2.4.1). When more than 2n parameters in (2.4.3) are used to specify the same I/O properties, we say that the plant is overparameterized. For example, the plant Z(s) Λ(s) y= u (2.4.5) R(s) Λ(s) where Λ(s) is Hurwitz of arbitrary degree r > 0, has the same I/O properties as the plant described by (2.4.3), and it is, therefore, overparameterized. In addition, any state representation of order n+r > n of (2.4.5) is nonminimal. For some estimation and control problems, certain plant parameterizations are more convenient than others. A plant parameterization that is useful for parameter estimation and some control problems is the one where parameters are lumped together and separated from signals. In parameter estimation, the parameters are the unknown constants to be estimated from the measurements of the I/O signals of the plant. In the following sections, we present various parameterizations of the same plant that are useful for parameter estimation to be studied in later chapters.

2.4. PLANT PARAMETRIC MODELS

2.4.1

49

Linear Parametric Models

Parameterization 1 The plant equation (2.4.3) may be expressed as an nth-order differential equation given by y (n) + an−1 y (n−1) + · · · + a0 y = bn−1 u(n−1) + bn−2 u(n−2) + · · · + b0 u (2.4.6) If we lump all the parameters in (2.4.6) in the parameter vector θ∗ = [bn−1 , bn−2 , . . . , b0 , an−1 , an−2 , . . . , a0 ]> and all I/O signals and their derivatives in the signal vector Y

= [u(n−1) , u(n−2) , . . . , u, −y (n−1) , −y (n−2) , . . . , −y]> > > = [αn−1 (s)u, −αn−1 (s)y]> 4

where αi (s) = [si , si−1 , . . . , 1]> , we can express (2.4.6) and, therefore, (2.4.3) in the compact form y (n) = θ∗> Y (2.4.7) Equation (2.4.7) is linear in θ∗ , which, as we show in Chapters 4 and 5, is crucial for designing parameter estimators to estimate θ∗ from the measurements of y (n) and Y . Because in most applications the only signals available for measurement is the input u and output y and the use of differentiation is not desirable, the use of the signals y (n) and Y should be avoided. One way to avoid them is to filter each side of (2.4.7) with an nth-order stable 1 filter Λ(s) to obtain z = θ∗> φ where 4

z= "

1 (n) sn y = y Λ(s) Λ(s)

> (s) αn−1 α> (s) φ= u, − n−1 y Λ(s) Λ(s) 4

(2.4.8)

#>

and Λ(s) = sn + λn−1 sn−1 + · · · + λ0

50

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS

is an arbitrary Hurwitz polynomial in s. It is clear that the scalar signal z and vector signal φ can be generated, without the use of differentiators, by simply si filtering the input u and output y with stable proper filters Λ(s) , i = 0, 1, . . . n. If we now express Λ(s) as Λ(s) = sn + λ> αn−1 (s) where λ = [λn−1 , λn−2 , . . . , λ0 ]> , we can write z=

sn Λ(s) − λ> αn−1 (s) αn−1 (s) y= y = y − λ> y Λ(s) Λ(s) Λ(s)

Therefore, y = z + λ>

αn−1 (s) y Λ(s)

Because z = θ∗> φ = θ1∗> φ1 + θ2∗> φ2 , where 4

4

θ1∗ = [bn−1 , bn−2 , . . . , b0 ]> , θ2∗ = [an−1 , an−2 , . . . , a0 ]> 4

φ1 =

αn−1 (s) αn−1 (s) 4 u, φ2 = − y Λ(s) Λ(s)

it follows that y = θ1∗> φ1 + θ2∗> φ2 − λ> φ2 Hence, y = θλ∗> φ

(2.4.9)

where θλ∗ = [θ1∗> , θ2∗> − λ> ]> . Equations (2.4.8) and (2.4.9) are represented by the block diagram shown in Figure 2.2. A state-space representation for generating the signals in (2.4.8) and (2.4.9) may be obtained by using the identity [adj(sI − Λc )]l = αn−1 (s) where Λc , l are given by   

Λc =   



−λn−1 −λn−2 · · · −λ0 1 0 ··· 0   .. ..  .. , . . .  0 ··· 1 0

  

l=  

1 0 .. . 0

     

2.4. PLANT PARAMETRIC MODELS u- αn−1 (s) φ1- ∗> θ1 Λ(s)

+ Σl + −6 AK A

51 yφ2 −αn−1 (s) ¾ θ2∗> ¾ Λ(s)

λ> ¾ + ? + - Σl zFigure 2.2 Plant Parameterization 1. which implies that det(sI − Λc ) = Λ(s), (sI − Λc )−1 l =

αn−1 (s) Λ(s)

Therefore, it follows from (2.4.8) and Figure 2.2 that φ˙ 1 = Λc φ1 + lu, φ˙ 2 = Λc φ2 − ly,

φ1 ∈ Rn φ2 ∈ Rn

y = θλ∗> φ

(2.4.10) >

z = y + λ φ2 = θ

∗>

φ

Because Λ(s) = det(sI − Λc ) and Λ(s) is Hurwitz, it follows that Λc is a stable matrix. The parametric model (2.4.10) is a nonminimal state-space representation of the plant (2.4.3). It is nonminimal because 2n integrators are used to represent an nth-order system. Indeed, the transfer function Y (s)/U (s) computed using (2.4.10) or Figure 2.2, i.e., Y (s) Z(s) Λ(s) Z(s) = = U (s) R(s) Λ(s) R(s) involves n stable zero-pole cancellations. The plant (2.4.10) has the same I/O response as (2.4.3) and (2.4.1) provided that all state initial conditions are equal to zero, i.e., x0 = 0, φ1 (0) = φ2 (0) = 0. In an actual plant, the state x in (2.4.1) may represent physical variables and the initial state x0 may be different from zero. The

52

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS

effect of the initial state x0 may be accounted for in the model (2.4.10) by applying the same procedure to equation (2.4.2) instead of equation (2.4.3). We can verify (see Problem 2.9) that if we consider the effect of initial condition x0 , we will obtain the following representation φ˙ 1 = Λc φ1 + lu, φ˙ 2 = Λc φ2 − ly, y =

θλ∗> φ

φ1 (0) = 0 φ2 (0) = 0

+ η0

(2.4.11)

z = y + λ> φ2 = θ∗> φ + η0 where η0 is the output of the system ω˙ = Λc ω, η0 =

ω(0) = ω0

C0> ω

(2.4.12)

where ω ∈ Rn , ω0 = B0 x0 and C0 ∈ Rn , B0 ∈ Rn×n are constant matrices that satisfy C0> {adj(sI − Λc )}B0 = C > {adj(sI − A)}. Because Λc is a stable matrix, it follows from (2.4.12) that ω, η0 converge to zero exponentially fast. Therefore, the effect of the nonzero initial condition x0 is the appearance of the exponentially decaying to zero term η0 in the output y and z. Parameterization 2 Let us now consider the parametric model (2.4.9) y = θλ∗> φ −1 (s) = 1, where W (s) = Z (s)/R (s) is a and the identity Wm (s)Wm m m m transfer function with relative degree one, and Zm (s) and Rm (s) are Hurwitz polynomials. Because θλ∗ is a constant vector, we can express (2.4.9) as −1 y = Wm (s)θλ∗> Wm (s)φ

If we let

"

> (s) αn−1 α> (s) 1 ψ= φ= u, − n−1 y Wm (s) Wm (s)Λ(s) Wm (s)Λ(s) 4

#>

2.4. PLANT PARAMETRIC MODELS u -

αn−1 (s) Λ(s)Wm (s)

ψ1-

θ1∗>

53

+ Σl

+ −6 AK A

- Wm (s)

ψ2 θ2∗> ¾

y-

−αn−1 (s) ¾ Λ(s)Wm (s)

λ> ¾

Figure 2.3 Plant Parameterization 2. we have y = Wm (s)θλ∗> ψ

(2.4.13)

αn−1 (s) are proper transfer functions with Because all the elements of Λ(s)W m (s) > > > stable poles, the state ψ = [ψ1 ,ψ2 ] , where

ψ1 =

αn−1 (s) αn−1 (s) u, ψ2 = − y Wm (s)Λ(s) Wm (s)Λ(s)

can be generated without differentiating y or u. The dimension of ψ depends on the order n of Λ(s) and the order of Zm (s). Because Zm (s) can be arbitrary, the dimension of ψ can be also arbitrary. Figure 2.3 shows the block diagram of the parameterization of the plant given by (2.4.13). We refer to (2.4.13) as Parameterization 2. In [201], Parameterization 2 is referred to as the model reference representation and is used to design parameter estimators for estimating θλ∗ when Wm (s) is a strictly positive real transfer function (see definition in Chapter 3). A special case of (2.4.13) is the one shown in Figure 2.4 where Wm (s) =

1 s + λ0

and (s + λ0 ) is a factor of Λ(s), i.e., Λ(s) = (s + λ0 )Λq (s) = sn + λn−1 sn−1 + · · · + λ0 where Λq (s) = sn−1 + qn−2 sn−2 + · · · + q1 s + 1

54

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS u -

αn−1 (s) Λq (s)

ψ1-

θ1∗>

+ Σl

+ −6 AK A

-

1 s+λ0

ψ2 θ2∗> ¾

y-

−αn−1 (s) ¾ Λq (s)

λ> ¾

Figure 2.4 Plant Parameterization 2 with Λ(s) = (s + λ0 )Λq (s) and 1 . Wm (s) = s+λ 0 The plant Parameterization 2 of Figure 2.4 was first suggested in [131], where it was used to develop stable adaptive observers. An alternative parametric model of the plant of Figure 2.4 can be obtained by first separating (s) the biproper elements of αΛn−1 as follows: q (s) 4

For any vector c = [cn−1 , cn−2 , . . . , c1 , c0 ]> ∈ Rn , we have cn−1 sn−1 c¯> αn−2 (s) c> αn−1 (s) = + Λq (s) Λq (s) Λq (s) 4

(2.4.14)

4

where c¯ = [cn−2 , . . . , c1 , c0 ]> , αn−2 = [sn−2 , . . . , s, 1]> . Because Λq (s) = sn−1 + q¯> αn−2 (s), where q¯ = [qn−2 , . . . , q1 , 1]> , we have sn−1 = Λq (s) − q¯> αn−2 , which after substitution we obtain c> αn−1 (s) (¯ c − cn−1 q¯)> αn−2 (s) = cn−1 + Λq (s) Λq (s)

(2.4.15)

We use (2.4.15) to obtain the following expressions: αn−1 (s) αn−2 (s) u = bn−1 u + θ¯1∗> u Λq (s) Λq (s) αn−1 (s) αn−2 (s) −(θ2∗> − λ> ) y = (λn−1 − an−1 )y − θ¯2∗> y (2.4.16) Λq (s) Λq (s) θ1∗>

¯ − (an−1 − λn−1 )¯ where θ¯1∗> = ¯b − bn−1 q¯, θ¯2∗> = a ¯−λ q and a ¯ = [an−2 , > > > ¯ ¯ . . ., a1 , a0 ] , b = [bn−1 , . . . , b1 , b0 ] , λ = [λn−2 , . . . , λ1 , λ0 ] . Using (2.4.16), Figure 2.4 can be reconfigured as shown in Figure 2.5.

2.4. PLANT PARAMETRIC MODELS

u -

αn−2 (s) Λq (s)

ψ¯1- bn−1

θ¯1∗>

55

+ l +Σ + ¢¸+6 AK ¢ A ¢ A

-

1 s + λ0

x ¯1 =-y

ψ¯2 −αn−2 (s) ¾ θ¯2∗> ¾ Λq (s)

λn−1 −an−1 ¾

Figure 2.5 Equivalent plant Parameterization 2. A nonminimal state space representation of the plant follows from Figure 2.5, i.e., ¯ x ¯˙ 1 = −λ0 x ¯1 + θ¯∗> ψ, ¯ c ψ¯1 + ¯lu, ψ˙¯1 = Λ ¯ c ψ¯2 − ¯ly, ψ¯˙ 2 = Λ

x ¯1 ∈ R1 ψ¯1 ∈ Rn−1 ψ¯2 ∈ Rn−1

(2.4.17)

y = x ¯1 where θ¯∗ = [bn−1 , θ¯1∗> , λn−1 − an−1 , θ¯2∗> ]> , ψ¯ = [u, ψ¯1> , y, ψ¯2> ]> and   

¯c =  Λ  



−qn−2 −qn−3 · · · −q0 1 0 ··· 0   .. ..  .. , . . .  0 ··· 1 0

  

¯l =   

1 0 .. .

     

0

As with Parameterization 1, if we account for the initial condition x(0) = x0 6= 0, we obtain ¯ x ¯˙ 1 = −λ0 x ¯1 + θ¯∗> ψ, ¯ c ψ¯1 + ¯lu, ψ˙¯1 = Λ ¯ c ψ¯2 − ¯ly, ψ¯˙ 2 = Λ

x ¯1 (0) = 0 ψ¯1 (0) = 0 ψ¯2 (0) = 0

y = x ¯ 1 + η0 where η0 is the output of the system ω˙ = Λc ω, η0 = C0> ω

ω(0) = ω0 , ω ∈ Rn

(2.4.18)

56

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS

where Λc , C0 , and ω0 are as defined in (2.4.12). Example 2.4.1 (Parameterization 1) Let us consider the differential equation y (4) + a2 y (2) + a0 y = b2 u(2) + b0 u

(2.4.19)

that describes the motion of the cart with the two pendulums considered in Examples 2.2.1, 2.3.1, where a2 = −1.1(α1 + α2 ), a0 = 1.2α1 α2 , b2 = β1 , b0 = −α1 β2 Equation (2.4.19) is of the same form as (2.4.6) with n = 4 and coefficients a3 = a1 = b3 = b1 = 0. Following (2.4.7), we may rewrite (2.4.19) in the compact form y (4) = θ0∗> Y0 θ0∗

>

(2)

(2.4.20)

(2)

>

where = [b2 , b0 , a2 , a0 ] , Y0 = [u , u, −y , −y] . Because y and u are the only signals we can measure, y (4) , Y0 are not available for measurement. 1 If we filter each side of (2.4.20) with the filter Λ(s) , where Λ(s) = (s + 2)4 = s4 + 8s3 + 24s2 + 32s + 16, we have z = θ0∗> φ0 4

h

2

(2.4.21) 2

i>

s 1 s 1 s are now signals where z = (s+2) 4 y, φ0 = (s+2)4 u, (s+2)4 u, − (s+2)4 y, − (s+2)4 y that can be generated from the measurements of y and u by filtering. Because in (2.4.19) the elements a3 = a1 = b3 = b1 = 0, the dimension of θ0∗ , φ0 is 4 instead of 8, which is implied by (2.4.8). Similarly, following (2.4.9) we have

y = θλ∗> φ where

(2.4.22)

θλ∗ = [0, b2 , 0, b0 , −8, a2 − 24, −32, a0 − 16]> · > ¸> α3 (s) α3> (s) φ= u, − y , α3 (s) = [s3 , s2 , s, 1]> (s + 2)4 (s + 2)4

elements of θλ∗ that do not depend on the parameters of (2.4.19) to obtain ∗> y = θ0λ φ 0 + h> 0φ ∗ where θ0λ = [b2 , b0 , a2 − 24, a0 − 16]> , h0 = [0, 0, 0, 0, −8, 0, −32, 0]> . We obtain a state-space representation of (2.4.21) and (2.4.22) by using (2.4.10), i.e.,

φ˙ 1 φ˙ 2 y

= = =

Λc φ1 + lu, φ1 ∈ R4 Λc φ2 − ly, φ2 ∈ R4 ∗> θλ∗> φ = θ0λ φ0 + h> 0φ

z

=

θ0∗> φ0

2.4. PLANT PARAMETRIC MODELS where



−8 −24  1 0 Λc =   0 1 0 0  0 1  0 0 φ0 =   0 0 0 0

57

 −32 −16 0 0  , 0 0  1 0 0 0 0 0

0 1 0 0

0 0 0 0

0 0 1 0



 1  0   l=  0  0  0 0  φ 0  1

0 0 0 0

> > and φ = [φ> 1 , φ2 ] . Instead of (2.4.22), we can also write

y = θ0∗> φ0 − λ> φ2 where λ = [8, 24, 32, 16]> .

5

Example 2.4.2 (Parameterization 2) Consider the same plant as in Example 2.4.1, i.e., y = θλ∗> φ where θλ∗> = [0, b2 , 0, b0 , −8, a2 − 24, −32, a0 − 16], ·

α3> (s) α3> (s) φ= u, − y (s + 2)4 (s + 2)4 Now we write y= where

· 4

ψ=

¸>

1 ∗> θ ψ s+2 λ

α3> (s) α> (s) u, − 3 y 3 (s + 2) (s + 2)3

Using simple algebra, we have  3   s  s2   α3 (s) 1  = = (s + 2)3 (s + 2)3  s   1

 1 1 0  + 0  (s + 2)3 0

¸>



−6  1   0 0

−12 0 1 0

 −8 0   α (s) 0  2 1

where α2 (s) = [s2 , s, 1]> . Therefore, ψ can be expressed as · ¸> α2> (s) α2> (s) > α2 (s) > α2 (s) ¯ ¯ ψ = u−λ u, u, −y + λ y, − y (s + 2)3 (s + 2)3 (s + 2)3 (s + 2)3

58

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS

¯ = [6, 12, 8]> , and θ∗> ψ can be expressed as where λ λ θλ∗> ψ = θ¯∗> ψ¯ where

(2.4.23)

θ¯∗ = [b2 , 0, b0 , 8, a2 + 24, 64, a0 + 48]> · > ¸> α2 (s) α2> (s) ¯ ψ= u, y, − y (s + 2)3 (s + 2)3

Therefore, y=

1 ¯∗> ¯ θ ψ s+2

(2.4.24)

A state-space realization of (2.4.24) is x ¯˙ 1 ψ˙¯

¯ = −2¯ x1 + θ¯∗> ψ, ¯ c ψ¯1 + ¯lu, = Λ

x ¯ 1 ∈ R1 ψ¯1 ∈ R3

ψ˙¯2 y

¯ c ψ¯2 − ¯ly, = Λ = x ¯1

ψ¯2 ∈ R3

1

where ψ¯ = [ψ¯1> , y, ψ¯2> ]> , 

−6 ¯c =  1 Λ 0

2.4.2

−12 0 1

   −8 1 0  , ¯l =  0  0 0

5

Bilinear Parametric Models

Let us now consider the parameterization of a special class of systems expressed as Z0 (s) y = k0 u (2.4.25) R0 (s) where k0 is a scalar, R0 (s) is monic of degree n, and Z0 (s) is monic and Hurwitz of degree m < n. In addition, let Z0 (s) and R0 (s) satisfy the Diophantine equation k0 Z0 (s)P (s) + R0 (s)Q(s) = Z0 (s)A(s) where Q(s) = sn−1 + q > αn−2 (s)

(2.4.26)

2.4. PLANT PARAMETRIC MODELS

59

P (s) = p> αn−1 (s) 4

αi (s) = [si , si−1 , . . . , s, 1]> q ∈ Rn−1 , p ∈ Rn are the coefficient vectors of Q(s) − sn−1 , P (s), respectively, and A(s) is a monic Hurwitz polynomial of degree 2n − m − 1. The Diophantine equation (2.4.26) relating Z0 (s), R0 (s), k0 to P (s), Q(s), and A(s) arises in control designs, such as model reference control, to be discussed in later chapters. The polynomials P (s) and Q(s) are usually the controller polynomials to be calculated by solving (2.4.26) for a given A(s). Our objective here is to obtain a parameterization of (2.4.25), in terms of the coefficients of P (s) and Q(s), that is independent of the coefficients of Z0 (s) and R0 (s). We achieve this objective by using (2.4.26) to eliminate the dependence of (2.4.25) on Z0 (s) and R0 (s) as follows: From (2.4.25), we obtain Q(s)R0 (s)y = k0 Z0 (s)Q(s)u

(2.4.27)

by rewriting (2.4.25) as R0 (s)y = k0 Z0 (s)u and operating on each side by Q(s). Using Q(s)R0 (s) = Z0 (s)(A(s) − k0 P (s)) obtained from (2.4.26) in (2.4.27), we have Z0 (s)(A(s) − k0 P (s))y = k0 Z0 (s)Q(s)u

(2.4.28)

Because Z0 (s) is Hurwitz, we filter each side of (2.4.28) by A(s)y = k0 P (s)y + k0 Q(s)u

1 Z0 (s)

to obtain (2.4.29)

and write (2.4.29) as A(s)y = k0 [p> αn−1 (s)y + q > αn−2 (s)u + sn−1 u]

(2.4.30)

We now have various choices to make. We can filter each side of (2.4.30) 1 with the stable filter A(s) and obtain "

y = k0

#

αn−1 αn−2 (s) sn−1 p> y + q> u+ u A(s) A(s) A(s)

which may be written in the compact form y = k0 (θ∗> φ + z0 )

(2.4.31)

60

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS ·

where

θ∗

=

[q > , p> ]> , φ

=

α> α> n−2 (s) n−1 (s) u, A(s) A(s) y

¸>

sn−1 A(s) u. We can 1 filter Λ(s) whose

, and z0 =

also filter each side of (2.4.30) using an arbitrary stable order nλ satisfies 2n − m − 1 ≥ nλ ≥ n − 1 to obtain y = W (s)k0 (θ∗> φ + z0 ) ·

where now φ =

α> α> n−1 (s) n−2 (s) u, Λ(s) Λ(s) y

¸>

, z0 =

sn−1 Λ(s) u,

(2.4.32) and W (s) =

Λ(s) A(s)

is a

proper transfer function. In (2.4.31) and (2.4.32), φ and z0 may be generated by filtering the input u and output y of the system. Therefore, if u and y are measurable, then all signals in (2.4.31) and (2.4.32) can be generated, and the only possible unknowns are k0 and θ∗ . If k0 is known, it can be absorbed in the signals φ and z0 , leading to models that are affine in θ∗ of the form y¯ = W (s)θ∗> φ¯

(2.4.33)

where y¯ = y − W (s)k0 z, φ¯ = k0 φ. If k0 , however, is unknown and is part of the parameters of interest, then (2.4.31) and (2.4.32) are not affine with respect to the parameters k0 and θ∗ , but instead, k0 and θ∗ appear in a special bilinear form. For this reason, we refer to (2.4.31) and (2.4.32) as bilinear parametric models to distinguish them from (2.4.7) to (2.4.9) and (2.4.33), which we refer to as linear parametric or affine parametric models. The forms of the linear and bilinear parametric models are general enough to include parameterizations of some systems with dynamics that are not necessarily linear, as illustrated by the following example. Example 2.4.3 Let us consider the nonlinear scalar system x˙ = a0 f (x, t) + b0 g(x, t) + c0 u

(2.4.34)

where a0 , b0 , and c0 are constant scalars; f (x, t) and g(x, t) are known nonlinear functions that can be calculated at each time t; and u, x is the input and state of the system, respectively. We assume that f, g, and u are such that for each initial condition x(0) = x0 , (2.4.34) has only one solution defined for all t ∈ [0, ∞). If x and u are measured, (2.4.34) can be expressed in the form of parametric model (2.4.33) by filtering each side of (2.4.34) with a stable strictly proper transfer function Wf (s), i.e., z = Wf (s)θ∗> φ (2.4.35)

2.5. PROBLEMS

61

where z = sWf (s)x, θ∗ = [a0 , b0 , c0 ]> , and φ = [f (x, t), g(x, t), u]> . Instead of (2.4.35), we may also write (2.4.34) in the form x˙ = −am x + am x + θ∗> φ for some am > 0, or x= Then 4

z =x−

1 [am x + θ∗> φ] s + am 1 am x= θ∗> φ s + am s + am

which is in the form of (2.4.35) with Wf (s) = (2.4.35) (respectively (2.4.36)) as z = θ∗> φf ,

1 s+am .

(2.4.36)

We may continue and rewrite

φf = Wf (s)φ

(2.4.37)

5

which is in the form of (2.4.8).

The nonlinear example demonstrates the fact that the parameter θ∗ appears linearly in (2.4.35) and (2.4.37) does not mean that the dynamics are linear.

2.5

Problems

2.1 Verify that x(t) and y(t) given by (2.2.4) satisfy the differential equation (2.2.2). 2.2 Check the controllability and observability of the following systems: (a) ·

−0.2 0 −1 0.8

x˙

=

y

= [−1, 1]x

¸

· x+

1 1

¸ u

(b)  x˙ = y

−1  0 0

1 −1 0

= [1, 1, 1]x

   0 0 0 x +  1 u −2 1

62

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS (c) · x˙ = y

=

−5 −6

1 0

¸

· x+

1 1

¸ u

[1, 1]x

2.3 Show that (A, B) is controllable if and only if the augmented matrix [sI −A, B] is of full rank for all s ∈ C. 2.4 The following state equation describes approximately the motion of a hot air balloon:          − τ11 0 0 0 x1 1 x˙ 1  x˙ 2  =  σ − τ12 0   x2  +  0  u +  τ12  w 0 x3 x˙ 3 0 0 1 0 y = [0 0 1]x where x1 : the temperature change of air in the balloon away from the equilibrium temperature; x2 : vertical velocity of the balloon; x3 : change in altitude from equilibrium altitude; u: control input that is proportional to the change in heat added to air in the balloon; w: vertical wind speed; and σ, τ1 , τ2 are parameters determined by the design of the balloon.

x1(temperature) u(hot air)

w (wind)

x3 (a) Let w = 0. Is the system completely controllable? Is it completely observable? (b) If it is completely controllable, transform the state-space representation into the controller canonical form. (c) If it is completely observable, transform the state-space representation into the observer canonical form.

2.5. PROBLEMS

63 4

(d) Assume w = constant. Can the augmented state xa = [x> , w]> be observed from y? (e) Assume u = 0. Can the states be controlled by w? 2.5 Derive the following transfer functions for the system described in Problem 2.4: 4 Y (s) U (s)

(a) G1 (s) =

when w = 0 and y = x3 .

4 Y (s) W (s)

when u = 0 and y = x3 .

4 Y1 (s) U (s)

when w = 0 and y1 = x1 .

4 Y1 (s) W (s)

when u = 0 and y1 = x1 .

(b) G2 (s) = (c) G3 (s) =

(d) G4 (s) =

2.6 Let a(s) = (s + α)3 , b(s) = β, where α, β are constants with β 6= 0. (a) Write the Sylvester matrix of a(s) and b(s). (b) Suppose p0 (s), l0 (s) is a solution of the polynomial equation a(s)l(s) + b(s)p(s) = 1

(2.5.1)

Show that (p1 (s), l1 (s)) is a solution of (2.5.1) if and only if p1 (s), l1 (s) can be expressed as p1 (s) = p0 (s) + r(s)a(s) l1 (s) = l0 (s) − r(s)b(s) for any polynomial r(s). (c) Find the solution of (2.5.1) for which p(s) has the lowest degree and p(s)/l(s) is a proper rational function. 2.7 Consider the third order plant y = G(s)u where G(s) =

b2 s2 + b1 s + b0 s3 + a2 s2 + a1 s + a0

(a) Write the parametric model of the plant in the form of (2.4.8) or (2.4.13) when θ∗ = [b2 , b1 , b0 , a2 , a1 , a0 ]> . (b) If a0 , a1 , and a2 are known, i.e., a0 = 2, a1 = 1, and a2 = 3, write a parametric model for the plant in terms of θ∗ = [b2 , b1 , b0 ]> . (c) If b0 , b1 , and b2 are known, i.e., b0 = 1, b1 = b2 = 0, develop a parametric model in terms of θ∗ = [a2 , a1 , a0 ]> .

64

CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS

2.8 Consider the spring-mass-dashpot system shown below:

u

x

M

k f

where k is the spring constant, f the viscous-friction or damping coefficient, m the mass of the system, u the forcing input, and x the displacement of the mass M . If we assume a “linear” spring, i.e., the force acting on the spring is proportional to the displacement, and a friction force proportional to velocity, i.e., x, ˙ we obtain, using Newton’s law, the differential equation Mx ¨ = u − kx − f x˙ that describes the dynamic system. (a) Give a state-space representation of the system. (b) Calculate the transfer function that relates x with u. (c) Obtain a linear parametric model of the form z = θ∗> φ where θ∗ = [M, k, f ]> and z, φ are signals that can be generated from the measurements of u, x without the use of differentiators. 2.9 Verify that (2.4.11) and (2.4.12) are nonminimal state-space representations of the system described by (2.4.1). Show that for the same input u(t), the output response y(t) is exactly the same for both systems. (Hint: Verify that C0> [adj(sI − Λc )]B0 = C > [adj(sI − A)] for some C0 ∈ Rn , B0 ∈ Rn×n by using the identity [adj(sI − A)] =

sn−1 I + sn−2 (A + an−1 I) + sn−3 (A2 + an−1 A + an−2 I) + · · · + (An−1 + an−1 An−2 + · · · + a1 I)

and choosing C0 such that (C0 , Λc ) is an observable pair.) 2.10 Write a state-space representation for the following systems:

2.5. PROBLEMS

65

(a) φ =

αn−1 (s) Λ(s) u,

Λ(s) is monic of order n.

(b) φ =

αn−1 (s) Λ1 (s) u,

Λ1 (s) is monic of order n − 1.

(c) φ =

αm (s) Λ1 (s) u,

2.11 Show that

m ≤ n − 1, Λ1 (s) is monic of order n − 1.

¢> ¡ αn−1 (s) (sI − Λc )−1 l = Co> (sI − Λo )−1 = Λ(s)

where (Λc , l) is in the controller form and (Co , Λo ) is in the observer form. 2.12 Show that there exists constant matrices Qi ∈ R(n−1)×(n−1) such that (sI − Λ0 )−1 di = Qi

αn−2 (s) , Λ(s)

i = 1, 2, . . . , n

¯  ¯ In−2 ¯ where d1 = −λ; Λ(s) = sn−1 + λ>αn−2 (s) = det(sI − Λ0 ), Λ0 = −λ ¯¯ −− ; ¯ 0 > n−1 di = [0, . . . , 0, 1, 0, . . . , 0] ∈ R whose (i − 1)th element is equal to 1, and i = 2, 3, . . . , n. 

Chapter 3

Stability 3.1

Introduction

The concept of stability is concerned with the investigation and characterization of the behavior of dynamic systems. Stability plays a crucial role in system theory and control engineering, and has been investigated extensively in the past century. Some of the most fundamental concepts of stability were introduced by the Russian mathematician and engineer Alexandr Lyapunov in [133]. The work of Lyapunov was extended and brought to the attention of the larger control engineering and applied mathematics community by LaSalle and Lefschetz [124, 125, 126], Krasovskii [107], Hahn [78], Massera [139], Malkin [134], Kalman and Bertram [97], and many others. In control systems, we are concerned with changing the properties of dynamic systems so that they can exhibit acceptable behavior when perturbed from their operating point by external forces. The purpose of this chapter is to present some basic definitions and results on stability that are useful for the design and analysis of control systems. Most of the results presented are general and can be found in standard textbooks. Others are more specific and are developed for adaptive systems. The proofs for most of the general results are omitted, and appropriate references are provided. Those that are very relevant to the understanding of the material presented in later chapters are given in detail. In Section 3.2, we present the definitions and properties of various norms 66

3.2. PRELIMINARIES

67

and functions that are used in the remainder of the book. The concept of I/O stability and some standard results from functional analysis are presented in Section 3.3. These include useful results on the I/O properties of linear systems, the small gain theorem that is widely used in robust control design and analysis, and the L2δ -norm and Bellman-Gronwall (B-G) Lemma that are important tools in the analysis of adaptive systems. The definitions of Lyapunov stability and related theorems for linear and nonlinear systems are presented in Section 3.4. The concept of passivity, in particular of positive real and strictly positive real transfer functions, and its relation to Lyapunov stability play an important role in the design of stable adaptive systems. Section 3.5 contains some basic results on positive real functions, and their connections to Lyapunov functions and stability that are relevant to adaptive systems. In Section 3.6, the focus is on some elementary results and principles that are used in the design and analysis of LTI feedback systems. We concentrate on the notion of internal stability that we use to motivate the correct way of computing the characteristic equation of a feedback system and determining its stability properties. The use of sensitivity and complementary sensitivity functions and some fundamental trade-offs in LTI feedback systems are briefly mentioned to refresh the memory of the reader. The internal model principle and its use to reject the effects of external disturbances in feedback systems is presented. A reader who is somewhat familiar with Lyapunov stability and the basic properties of norms may skip this chapter. He or she may use it as reference and come back to it whenever necessary. For the reader who is unfamiliar with Lyapunov stability and I/O properties of linear systems, the chapter offers a complete tutorial coverage of all the notions and results that are relevant to the understanding of the rest of the book.

3.2 3.2.1

Preliminaries Norms and Lp Spaces

For many of the arguments for scalar equations to be extended and remain valid for vector equations, we need an analog for vectors of the absolute value of a scalar. This is provided by the norm of a vector.

68

CHAPTER 3. STABILITY

Definition 3.2.1 The norm |x| of a vector x is a real valued function with the following properties: (i) |x| ≥ 0 with |x| = 0 if and only if x = 0 (ii) |αx| = |α||x| for any scalar α (iii) |x + y| ≤ |x| + |y| (triangle inequality) The norm |x| of a vector x can be thought of as the size or length of the vector x. Similarly, |x − y| can be thought of as the distance between the vectors x and y. An m × n matrix A represents a linear mapping from n-dimensional space Rn into m-dimensional space Rm . We define the induced norm of A as follows: Definition 3.2.2 Let | · | be a given vector norm. Then for each matrix A ∈ Rm×n , the quantity kAk defined by 4

kAk = sup x6=0

x∈Rn

|Ax| = sup |Ax| = sup |Ax| |x| |x|≤1 |x|=1

is called the induced (matrix) norm of A corresponding to the vector norm | · |. The induced matrix norm satisfies the properties (i) to (iii) of Definition 3.2.1. Some of the properties of the induced norm that we will often use in this book are summarized as follows: (i) |Ax| ≤ kAk|x|, ∀x ∈ Rn (ii) kA + Bk ≤ kAk + kBk (iii) kABk ≤ kAkkBk where A, B are arbitrary matrices of compatible dimensions. Table 3.1 shows some of the most commonly used norms on Rn . 4 It should be noted that the function kAks = maxij |aij |, where A ∈ Rm×n and aij is the (i, j) element of A satisfies the properties (i) to (iii) of Definition 3.2.1. It is not, however, an induced matrix norm because no vector norm exists such that k · ks is the corresponding induced norm. Note that k · ks does not satisfy property (c).

3.2. PRELIMINARIES

69

Table 3.1 Commonly used norms Norm on

Rn

Induced norm on Rm×n

|x|∞= maxi |xi | (infinity norm) |x|1 =

P

i |xi |

kAk1 = maxj

P

|x|2 = (

kAk∞ = maxi

2 1/2 i |xi | )

P

P

j

|aij | (row sum)

i |aij |

(column sum)

kAk2 = [λm (A> A)]1/2 , where λm (M )

(Euclidean norm)

is the maximum eigenvalue of M >

Example 3.2.1 (i) Let x = [1, 2, −10, 0] . Using Table 3.1, we have √ |x|∞ = 10, |x|1 = 13, |x|2 = 105 (ii) Let



0 A= 1 0

 5 0 , −10

· B=

−1 5 0 2

¸

Using Table 3.1, we have kAk1 = 15, kBk1 = 7, kABk1 = 35,

kAk2 = 11.18, kBk2 = 5.465, kABk2 = 22.91,

kAk∞ = 10 kBk∞ = 6 kABk∞ = 20

which can be used to verify property (iii) of the induced norm.

5

For functions of time, we define the Lp norm 4

kxkp =

µZ ∞ 0

|x(τ )|p dτ

¶1/p

for p ∈ [1, ∞) and say that x ∈ Lp when kxkp exists (i.e., when kxkp is finite). The L∞ norm is defined as 4

kxk∞ = sup |x(t)| t≥0

and we say that x ∈ L∞ when kxk∞ exists. In the above Lp , L∞ norm definitions, x(t) can be a scalar or a vector function. If x is a scalar function, then | · | denotes the absolute value. If x is a vector function in Rn then | · | denotes any norm in Rn .

70

CHAPTER 3. STABILITY Similarly, for sequences we define the lp norm as 4

kxkp =

Ã∞ X

!1/p p

|xi |

,

1≤p<∞

i=1

and the l∞ norm as

4

kxk∞ = sup |xi | i≥1

where x = (x1 , x2 , . . .) and xi ∈ R. We say x ∈ lp (respectively x ∈ l∞ ) if kxkp (respectively kxk∞ ) exists. We are usually concerned with classes of functions of time that do not belong to Lp . To handle such functions we define the Lpe norm 4

kxt kp =

µZ t 0

¶1 p

|x(τ )| dτ

p

for p ∈ [1, ∞) and say that x ∈ Lpe when kxt kp exists for any finite t. Similarly, the L∞e norm is defined as 4

kxt k∞ = sup |x(τ )| 0≤τ ≤t

The function t2 does not belong to Lp but t2 ∈ Lpe . Similarly, any continuous function of time belongs to Lpe but it may not belong to Lp . For each p ∈ [1, ∞], the set of functions that belong to Lp (respectively Lpe ) form a linear vector space called Lp space (respectively Lpe space) [42]. If we define the truncated function ft as ( 4

ft (τ ) =

f (τ ) 0 ≤ τ ≤ t 0 τ >t

for all t ∈ [0, ∞), then it is clear that for any p ∈ [1, ∞], f ∈ Lpe implies that ft ∈ Lp for any finite t. The Lpe space is called the extended Lp space and is defined as the set of all functions f such that ft ∈ Lp . It can be easily verified that the Lp and Lpe norms satisfy the properties of the norm given by Definition 3.2.1. It should be understood, however, that elements of Lp and Lpe are equivalent classes [42], i.e., if f, g ∈ Lp and kf − gkp = 0, the functions f and g are considered to be the same element of Lp even though f (t) 6= g(t) for some values of t. The following lemmas give some of the properties of Lp and Lpe spaces that we use later.

3.2. PRELIMINARIES

71 1 p

older’s Inequality) If p, q ∈ [1, ∞] and Lemma 3.2.1 (H¨ f ∈ Lp , g ∈ Lq imply that f g ∈ L1 and

+ 1q = 1, then

kf gk1 ≤ kf kp kgkq When p = q = 2, the H¨older’s inequality becomes the Schwartz inequality, i.e., kf gk1 ≤ kf k2 kgk2 (3.2.1) Lemma 3.2.2 (Minkowski Inequality) For p ∈ [1, ∞], f, g ∈ Lp imply that f + g ∈ Lp and kf + gkp ≤ kf kp + kgkp (3.2.2) The proofs of Lemma 3.2.1 and 3.2.2 can be found in any standard book on real analysis such as [199, 200]. We should note that the above lemmas also hold for the truncated functions ft , gt of f, g, respectively, provided f, g ∈ Lpe . For example, if f and g are continuous functions, then f, g ∈ Lpe , i.e., ft , gt ∈ Lp for any finite t ∈ [0, ∞) and from (3.2.1) we have k(f g)t k1 ≤ kft k2 kgt k2 , i.e., Z t 0

µZ t

|f (τ )g(τ )|dτ ≤

0

2

|f (τ )| dτ

¶ 1 µZ t 2 0

¶1 2

|g(τ )| dτ

2

(3.2.3)

which holds for any finite t ≥ 0. We use the above Schwartz inequality extensively throughout this book. Example 3.2.2 Consider the function f (t) =

1 1+t .

Then,

¯ ¯ ¶ 12 µZ ∞ ¯ 1 ¯ 1 ¯ = 1, kf k2 = =1 kf k∞ = sup ¯¯ dt ¯ (1 + t)2 t≥0 1 + t 0 Z kf k1 = Hence, f ∈ L2 t ≥ 0, we have

T

0

∞

1 dt = lim ln(1 + t) → ∞ t→∞ 1+t

L∞ but f 6∈ L1 ; f , however, belongs to L1e , i.e., for any finite Z

t 0

1 dτ = ln(1 + t) < ∞ 1+τ

5

72

CHAPTER 3. STABILITY

Example 3.2.3 Consider the functions f (t) = 1 + t, g(t) =

1 , 1+t

f or t ≥ 0

It is clear that f 6∈ Lp for any p ∈ [1, ∞] and g 6∈ L1 . Both functions, however, belong to Lpe ; and can be used to verify the Schwartz inequality (3.2.3) k(f g)t k1 ≤ kft k2 kgt k2 i.e.,

Z

µZ

t

t

1dτ ≤ 0

2

¶ 12 µZ

t

(1 + τ ) dτ 0

0

1 dτ (1 + τ )2

¶ 12

for any t ∈ [0, ∞) or equivalently µ t≤

t(t2 + 3t + 3) 3

which is true for any t ≥ 0.

¶ 12 µ

t 1+t

¶ 12

5

In the remaining chapters of the book, we adopt the following notation regarding norms unless stated otherwise. We will drop the subscript 2 from | · |2 , k · k2 when dealing with the Euclidean norm, the induced Euclidean norm, and the L2 norm. If x : R+ 7→ Rn , then |x(t)| represents the vector norm in Rn at each time t kxt kp represents the Lpe norm of the function |x(t)| kxkp represents the Lp norm of the function |x(t)| If A ∈ Rm×n , then kAki represents the induced matrix norm corresponding to the vector norm | · |i . If A : R+ 7→ Rm×n has elements that are functions of time t, then kA(t)ki represents the induced matrix norm corresponding to the vector norm | · |i at time t.

3.2.2

Properties of Functions

Let us start with some definitions. Definition 3.2.3 (Continuity) A function f : [0, ∞) 7→ R is continuous on [0, ∞) if for any given ²0 > 0 there exists a δ(²0 , t0 ) such that ∀t0 , t ∈ [0, ∞) for which |t − t0 | < δ(²0 , t0 ) we have |f (t) − f (t0 )| < ²0 .

3.2. PRELIMINARIES

73

Definition 3.2.4 (Uniform Continuity) A function f : [0, ∞) 7→ R is uniformly continuous on [0, ∞) if for any given ²0 > 0 there exists a δ(²0 ) such that ∀t0 , t ∈ [0, ∞) for which |t − t0 | < δ(²0 ) we have |f (t) − f (t0 )| < ²0 . Definition 3.2.5 (Piecewise Continuity) A function f : [0, ∞) 7→ R is piecewise continuous on [0, ∞) if f is continuous on any finite interval [t0 , t1 ] ⊂ [0, ∞) except for a finite number of points. Definition 3.2.6 (Absolute Continuity) A function f : [a, b] 7→ R is absolutely continuous on [a, b] iff, for any ²0 > 0, there is a δ > 0 such that n X

|f (αi ) − f (βi )| < ²0

i=1

for any finite collection of subintervals (αi , βi ) of [a, b] with

Pn

i=1 |αi −βi |

< δ.

Definition 3.2.7 (Lipschitz) A function f : [a, b] → R is Lipschitz on [a, b] if |f (x1 ) − f (x2 )| ≤ k|x1 − x2 | ∀x1 , x2 ∈ [a, b], where k ≥ 0 is a constant referred to as the Lipschitz constant. The function f (t) = sin( 1t ) is continuous on (0, ∞), but is not uniformly continuous (verify!). A function defined by a square wave of finite frequency is not continuous on [0, ∞), but it is piecewise continuous. Note that a uniformly continuous function is also continuous. A function f with f˙ ∈ L∞ is uniformly continuous on [0, ∞). Therefore, an easy way of checking the uniform continuity of f (t) is to check the boundedness of f˙. If f is Lipschitz on [a, b], then it is absolutely continuous. The following facts about functions are important in understanding some of the stability arguments which are often made in the analysis of adaptive systems. Fact 1 limt→∞ f˙(t) = 0 does not imply that f (t) has a limit as t → ∞. √ For example, consider the function f (t) = sin( 1 + t). We have √ cos 1 + t ˙ → 0 as t → ∞ f= √ 2 1+t

74

CHAPTER 3. STABILITY

but f (t) has no limit. Another example is f (t) = is an unbounded function of time. Yet

√ 1 + t sin(ln(1 + t)), which

sin(ln(1 + t)) cos(ln(1 + t)) √ √ f˙(t) = + → 0 as t → ∞ 2 1+t 1+t Fact 2 limt→∞ f (t) = c for some constant c ∈ R does not imply that f˙(t) → 0 as t → ∞. For example, the function f (t) = finite integer n but

sin(1+t)n 1+t

tends to zero as t → ∞ for any

n

sin(1 + t) + n(1 + t)n−2 cos(1 + t)n f˙ = − (1 + t)2 has no limit for n ≥ 2 and becomes unbounded as t → ∞ for n > 2. Some important lemmas that we frequently use in the analysis of adaptive schemes are the following: Lemma 3.2.3 The following is true for scalar-valued functions: (i) A function f (t) that is bounded from below and is nonincreasing has a limit as t → ∞. (ii) Consider the nonnegative scalar functions f (t), g(t) defined for all t ≥ 0. If f (t) ≤ g(t) ∀t ≥ 0 and g ∈ Lp , then f ∈ Lp for all p ∈ [1, ∞]. Proof (i) Because f is bounded from below, its infimum fm exists, i.e., fm =

inf

0≤t≤∞

f (t)

which implies that there exists a sequence {tn } ∈ R+ such that limn→∞ f (tn ) = fm . This, in turn, implies that given any ²0 > 0 there exists an integer N > 0 such that |f (tn ) − fm | < ²0 ,

∀n ≥ N

Because f is nonincreasing, there exists an n0 ≥ N such that for any t ≥ tn0 and some n0 ≥ N we have f (t) ≤ f (tn0 ) and |f (t) − fm | ≤ |f (tn0 ) − fm | < ²0 for any t ≥ tn0 . Because ²0 > 0 is any given number, it follows that limt→∞ f (t) = fm .

3.2. PRELIMINARIES

75

(ii) We have µZ 4

z(t) =

t

¶ p1

p

µZ

f (τ )dτ

∞

≤

¶ p1

p

g (τ )dτ

0

< ∞, ∀t ≥ 0

0

Because 0 ≤ z(t) < ∞ and z(t) is nondecreasing, we can establish, as in (i), that z(t) has a limit, i.e., limt→∞ z(t) = z¯ < ∞, which implies that f ∈ Lp . For p = ∞, the proof is straightforward. 2

Lemma 3.2.3 (i) does not imply that f ∈ L∞ . For example, the function f (t) = 1t with t ∈ (0, ∞) is bounded from below, i.e., f (t) ≥ 0 and is nonincreasing, but it becomes unbounded as t → 0. If, however, f (0) is finite, then it follows from the nonincreasing property f (t) ≤ f (0) ∀t ≥ 0 that f ∈ L∞ . A special case of Lemma 3.2.3 that we often use in this book is when f ≥ 0 and f˙ ≤ 0. Lemma 3.2.4 Let f, V : [0, ∞) 7→ R. Then V˙ ≤ −αV + f,

∀t ≥ t0 ≥ 0

implies that V (t) ≤ e

−α(t−t0 )

Z t

V (t0 ) +

t0

e−α(t−τ ) f (τ )dτ,

∀t ≥ t0 ≥ 0

for any finite constant α. 4 Proof Let w(t) = V˙ + αV − f . We have w(t) ≤ 0 and

V˙ = −αV + f + w implies that Z −α(t−t0 )

V (t) = e

Z

t

V (t0 ) +

e

−α(t−τ )

t

f (τ )dτ +

t0

e−α(t−τ ) w(τ )dτ

t0

Because w(t) ≤ 0 ∀t ≥ t0 ≥ 0, we have Z V (t) ≤ e

−α(t−t0 )

t

V (t0 ) +

e−α(t−τ ) f (τ )dτ

t0

2

76

CHAPTER 3. STABILITY

Lemma 3.2.5 If f, f˙ ∈ L∞ and f ∈ Lp for some p ∈ [1, ∞), then f (t) → 0 as t → ∞. The result of Lemma 3.2.5 is a special case of a more general result given by Barb˘alat’s Lemma stated below. R

Lemma 3.2.6 (Barb˘ alat’s Lemma [192]) If limt→∞ 0t f (τ )dτ exists and is finite, and f (t) is a uniformly continuous function, then limt→∞ f (t) = 0. Proof Assume that limt→∞ f (t) = 0 does not hold, i.e., either the limit does not exist or it is not equal to zero. This implies that there exists an ²0 > 0 such that for every T > 0, one can find a sequence of numbers ti > T such that |f (ti )| > ²0 for all i. Because f is uniformly continuous, there exists a number δ(²0 ) > 0 such that |f (t) − f (ti )| <

²0 for every t ∈ [ti , ti + δ(²0 )] 2

Hence, for every t ∈ [ti , ti + δ(²0 )], we have |f (t)|

= |f (t) − f (ti ) + f (ti )| ≥ |f (ti )| − |f (t) − f (ti )| ²0 ²0 ≥ ²0 − = 2 2

which implies that ¯Z ¯ Z ¯ ti +δ(²0 ) ¯ ti +δ(²0 ) ²0 δ(²0 ) ¯ ¯ f (τ )dτ ¯ = |f (τ )| dτ > ¯ ¯ ti ¯ 2 ti

(3.2.4)

where the first equality holds because f (t) retains the same sign for t ∈ [ti , ti +δ(²0 )]. 4 Rt On the other hand, g(t) = 0 f (τ )dτ has a limit as t → ∞ implies that there exists a T (²0 ) > 0 such that for any t2 > t1 > T (²0 ) we have |g(t1 ) − g(t2 )| < i.e.,

¯Z ¯ ¯ ¯

t2

t1

²0 δ(²0 ) 2

¯ ¯ ²0 δ(²0 ) f (τ )dτ ¯¯ < 2

which for t2 = ti +δ(²0 ), t1 = ti contradicts (3.2.4), and, therefore, limt→∞ f (t) = 0.

2

3.2. PRELIMINARIES

77

The proof of Lemma 3.2.5 follows directly from that of Lemma 3.2.6 by noting that the function f p (t) is uniformly continuous for any p ∈ [1, ∞) because f, f˙ ∈ L∞ . The condition that f (t) is uniformly continuous is crucial for the results of Lemma 3.2.6 to hold as demonstrated by the following example. Example 3.2.4 Consider the following function described by a sequence of isosceles triangles of base length n12 and height equal to 1 centered at n where n = 1, 2, . . . ∞ as shown in the figure below:

1

f (t) 6 ¡@

¡

¡ ¾

¤C ¤ C

@

1 2

@ -

¤C ¤ C

······

¤ C ¾2-

¤ C n ¾ -

1 4

······ -

1 n2

This function is continuous but not uniformly continuous. It satisfies Z

t

lim

t→∞

f (τ )dτ = 0

∞ π2 1X 1 = 2 2 n=1 n 12

5

but limt→∞ f (t) does not exist.

The above example also serves as a counter example to the following situation that arises in the analysis of adaptive systems: We have a function V (t) with the following properties: V (t) ≥ 0, V˙ ≤ 0. As shown by Lemma 3.2.3 these properties imply that limt→∞ V (t) = V∞ exists. However, there is no guarantee that V˙ (t) → 0 as t → ∞. For example consider the function V (t) = π −

Z t 0

f (τ )dτ

where f (t) is as defined in Example 3.2.4. Clearly, V (t) ≥ 0,

V˙ = −f (t) ≤ 0,

and lim V (t) = V∞ = π −

t→∞

∀t ≥ 0 π2 12

78

CHAPTER 3. STABILITY

but limt→∞ V˙ (t) = − limt→∞ f (t) does not exist. According to Barb˘alat’s lemma, a sufficient condition for V˙ (t) → 0 as t → ∞ is that V˙ is uniformly continuous.

3.2.3

Positive Definite Matrices

A square matrix A ∈ Rn×n is called symmetric if A = A> . A symmetric matrix A is called positive semidefinite if for every x ∈ Rn , x> Ax ≥ 0 and positive definite if x> Ax > 0 ∀x ∈ Rn with |x| 6= 0. It is called negative semidefinite (negative definite) if −A is positive semidefinite (positive definite). The definition of a positive definite matrix can be generalized to nonsymmetric matrices. In this book we will always assume that the matrix is symmetric when we consider positive or negative definite or semidefinite properties. We write A ≥ 0 if A is positive semidefinite, and A > 0 if A is positive definite. We write A ≥ B and A > B if A − B ≥ 0 and A − B > 0, respectively. A symmetric matrix A ∈ Rn×n is positive definite if and only if any one of the following conditions holds: (i) λi (A) > 0, i = 1, 2, . . . , n where λi (A) denotes the ith eigenvalue of A, which is real because A = A> . (ii) There exists a nonsingular matrix A1 such that A = A1 A> 1. (iii) Every principal minor of A is positive. (iv) x> Ax ≥ α|x|2 for some α > 0 and ∀x ∈ Rn . The decomposition A = A1 A> 1 in (ii) is unique when A1 is also symmetric. In this case, A1 is positive definite, it has the same eigenvectors as A, and its eigenvalues are equal to the square roots of the corresponding eigenvalues 1 of A. We specify this unique decomposition of A by denoting A1 as A 2 , i.e., 1 > 1 A = A 2 A 2 where A 2 is a positive definite matrix and A>/2 denotes the transpose of A1/2 . A symmetric matrix A ∈ Rn×n has n orthogonal eigenvectors and can be decomposed as A = U > ΛU (3.2.5) where U is a unitary (orthogonal) matrix (i.e., U > U = I) with the eigen-

3.3. INPUT/OUTPUT STABILITY

79

vectors of A, and Λ is a diagonal matrix composed of the eigenvalues of A. Using (3.2.5), it follows that if A ≥ 0, then for any vector x ∈ Rn λmin (A)|x|2 ≤ x> Ax ≤ λmax (A)|x|2 Furthermore, if A ≥ 0 then kAk2 = λmax (A) and if A > 0 we also have kA−1 k2 =

1 λmin (A)

where λmax (A), λmin (A) is the maximum and minimum eigenvalue of A, respectively. We should note that if A > 0 and B ≥ 0, then A + B > 0, but it is not true in general that AB ≥ 0.

3.3

Input/Output Stability

The systems encountered in this book can be described by an I/O mapping that assigns to each input a corresponding output, or by a state variable representation. In this section we shall present some basic results concerning I/O stability. These results are based on techniques from functional analysis [42], and most of them can be applied to both continuous- and discretetime systems. Similar results are developed in Section 3.4 by using the state variable approach and Lyapunov theory.

3.3.1

Lp Stability

We consider an LTI system described by the convolution of two functions u, h : R+ → R defined as 4

y(t) = u ∗ h =

Z t 0

h(t − τ )u(τ )dτ =

Z t 0

u(t − τ )h(τ )dτ

(3.3.1)

where u, y is the input and output of the system, respectively. Let H(s) be the Laplace transform of the I/O operator h(·). H(s) is called the transfer

80

CHAPTER 3. STABILITY

function and h(t) the impulse response of the system (3.3.1). The system (3.3.1) may also be represented in the form Y (s) = H(s)U (s)

(3.3.2)

where Y (s), U (s) is the Laplace transform of y, u respectively. We say that the system represented by (3.3.1) or (3.3.2) is Lp stable if u ∈ Lp ⇒ y ∈ Lp and kykp ≤ ckukp for some constant c ≥ 0 and any u ∈ Lp . When p = ∞, Lp stability, i.e., L∞ stability, is also referred to as bounded-input bounded-output (BIBO) stability. The following results hold for the system (3.3.1). Theorem 3.3.1 If u ∈ Lp and h ∈ L1 then kykp ≤ khk1 kukp

(3.3.3)

where p ∈ [1, ∞]. When p = 2 we have a sharper bound for kykp than that of (3.3.3) given by the following Lemma. Lemma 3.3.1 If u ∈ L2 and h ∈ L1 , then kyk2 ≤ sup |H(jω)|kuk2 ω

(3.3.4)

For the proofs of Theorem 3.3.1, Lemma 3.3.1 see [42]. Remark 3.3.1 It can be shown that (3.3.4) also holds [232] when h(·) is of the form ( 0 t<0 h(t) = P∞ i=0 fi δ(t − ti ) + fa (t) t ≥ 0 P

where fa ∈ L1 , ∞ i=0 |fi | < ∞ and ti are nonnegative finite constants. The Laplace transform of h(t) is now given by H(s) =

∞ X

fi e−sti + Ha (s)

i=0

which is not a rational function of s. The biproper transfer functions that are of interest in this book belong to the above class.

3.3. INPUT/OUTPUT STABILITY

81

Remark 3.3.2 We should also note that (3.3.3) and (3.3.4) hold for the truncated functions of u, y, i.e., kyt kp ≤ khk1 kut kp for any t ∈ [0, ∞) provided u ∈ Lpe . Similarly, kyt k2 ≤ sup |H(jω)|kut k2 ω

for any t ∈ [0, ∞) provided u ∈ L2e . This is clearly seen by noticing that u ∈ Lpe ⇒ ut ∈ Lp for any finite t ≥ 0.

It can be shown [42] that inequality (3.3.3) is sharp for p = ∞ because 4

khk1 is the induced norm of the map T : u 7→ T u = y from L∞ into L∞ , i.e., kT k∞ = khk1 . Similarly for (3.3.4) it can be shown that the induced norm of the linear map T : L2 7→ L2 is given by kT k2 = sup |H(jω)|

(3.3.5)

ω∈R

i.e., the bound (3.3.4) is also sharp. The induced L2 norm in (3.3.5) is referred to as the H∞ norm for the transfer function H(s) and is denoted by 4

kH(s)k∞ = sup |H(jω)| ω∈R

Let us consider the simple case where h(t) in (3.3.1) is the impulse response of an LTI system whose transfer function H(s) is a rational function of s. The following theorem and corollaries hold. Theorem 3.3.2 Let H(s) be a strictly proper rational function of s. Then H(s) is analytic in Re[s] ≥ 0 if and only if h ∈ L1 . Corollary 3.3.1 If h ∈ L1 , then (i) h decays exponentially , i.e., |h(t)| ≤ α1 e−α0 t for some α1 , α0 > 0 T (ii) u ∈ L1 ⇒ y ∈ L1 L∞ , y˙ ∈ L1 , y is continuous and limt→∞ |y(t)| = 0 T (iii) u ∈ L2 ⇒ y ∈ L2 L∞ , y˙ ∈ L2 , y is continuous and limt→∞ |y(t)| = 0 (iv) For p ∈ [1, ∞], u ∈ Lp ⇒ y, y˙ ∈ Lp and y is continuous

82

CHAPTER 3. STABILITY

For proofs of Theorem 3.3.2 and Corollary 3.3.1, see [42]. Corollary 3.3.2 Let H(s) be biproper and analytic in Re[s] ≥ 0. Then u ∈ T T L2 L∞ and limt→∞ |u(t)| = 0 imply that y ∈ L2 L∞ and limt→∞ |y(t)| = 0. Proof H(s) may be expressed as H(s) = d + Ha (s) where d is a constant and Ha (s) is strictly proper and analytic in Re[s] ≥ 0. We have y = du + ya , ya = Ha (s)u T where, by Corollary 3.3.1, ya ∈ L2 L∞ and |ya (t)| → 0 as t → ∞. Because T T u ∈ L2 L∞ and u(t) → 0 as t → ∞, it follows that y ∈ L2 L∞ and |y(t)| → 0 as t → ∞. 2 Example 3.3.1 Consider the system described by y = H(s)u,

H(s) =

e−αs s+β

for some constant α > 0. For β > 0, H(s) is analytic in Re[s] ≥ 0. The impulse response of the system is given by ½ −β(t−α) e t≥α h(t) = 0 t<α and h ∈ L1 if and only if β > 0. We have Z ∞ Z khk1 = |h(t)|dt = 0

and kH(s)k∞

∞

e−β(t−α) dt =

α

1 β

¯ −αjω ¯ ¯e ¯ ¯= 1 = sup ¯¯ jω + β ¯ β ω

5 Example 3.3.2 Consider the system described by y = H(s)u,

H(s) =

2s + 1 s+5

3.3. INPUT/OUTPUT STABILITY The impulse response of the system is given by ½ 2δ∆ (t) − 9e−5t h(t) = 0

83

t≥0 t<0

where ha = −9e−5t ∈ L1 . This system belongs to the class described in Remark 3.3.1. We have ¯ ¯ µ ¶1 ¯ 1 + 2jω ¯ 1 + 4ω 2 2 ¯ ¯ = sup kH(s)k∞ = sup ¯ =2 5 + jω ¯ 25 + ω 2 ω ω Hence, according to (3.3.4) and Remarks 3.3.1 and 3.3.2, for any u ∈ L2e , we have kyt k2 ≤ 2kut k2

5

for any t ∈ [0, ∞).

Definition 3.3.1 (µ−small in the mean square sense (m.s.s.)) Let x : [0, ∞) 7→ Rn , where x ∈ L2e , and consider the set (

S(µ) =

¯Z ) ¯ t+T ¯ x : [0, ∞) 7→ Rn ¯ x> (τ )x(τ )dτ ≤ c0 µT + c1 , ∀t, T ≥ 0 ¯ t

for a given constant µ ≥ 0, where c0 , c1 ≥ 0 are some finite constants, and c0 is independent of µ. We say that x is µ−small in the m.s.s. if x ∈ S(µ). Using the proceeding definition, we can obtain a result similar to that of Corollary 3.3.1 (iii) in the case where u ∈ / L2 but u ∈ S(µ) for some constant µ ≥ 0. Corollary 3.3.3 Consider the system (3.3.1). If h ∈ L1 , then u ∈ S(µ) implies that y ∈ S(µ) and y ∈ L∞ for any finite µ ≥ 0. Furthermore |y(t)|2 ≤

α12 eα 0 (c0 µ + c1 ), α0 (1 − e−α0 )

∀t ≥ t0 ≥ 0

where α0 , α1 are the parameters in the bound for h in Corollary 3.3.1 (i). Proof Using Corollary 3.3.1 (i), we have Z t Z t |y(t)| ≤ |h(t − τ )u(τ )|dτ ≤ α1 e−α0 (t−τ ) |u(τ )|dτ, t0

t0

∀t ≥ t0 ≥ 0

84

CHAPTER 3. STABILITY

for some constants α1 , α0 > 0. Using the Schwartz inequality we obtain Z t Z t |y(t)|2 ≤ α12 e−α0 (t−τ ) dτ e−α0 (t−τ ) |u(τ )|2 dτ t0

Z

α12 α0

≤

t0

t

e−α0 (t−τ ) |u(τ )|2 dτ

(3.3.6)

t0

Therefore, for any t ≥ t0 ≥ 0 and T ≥ 0 we have Z

t+T

2

|y(τ )| dτ

α12 α0 2 Z

≤

t

α1 α0

=

Z

Z

t+T

τ

e−α0 (τ −s) |u(s)|2 dsdτ

t t0 t+TµZ t −α0 (τ −s)

e

t

Z |u(s)|2 ds +

t0

¶ e−α0 (τ −s) |u(s)|2 ds dτ

τ

t

(3.3.7) Using the identity involving the change of the sequence of integration, i.e., Z

Z

t+T

Z

τ

f (τ ) t

t

Z

t+T

g(s)dsdτ =

t+T

g(s)

f (τ )dτ ds

t

(3.3.8)

s

for the second term on the right-hand side of (3.3.7), we have Z

t+T

2

|y(τ )| dτ

≤

t

α12 α0 +

Z

α12

Z

t+T

e Z

≤

t

dτ

t

t0 t+T

e

α0 t ÃZ t 2

α1 α02

−α0 τ

|u(s)|

t+T

2

e

!

−α0 τ

dτ

ds

s

Z e

t0

α0 s

eα0 s |u(s)|2 ds ÃZ

−α0 (t−s)

2

t+T

|u(s)| ds +

! 2

|u(s)| ds t

where the last inequality is obtained by using e−α0 t − e−α0 (t+T ) ≤ e−α0 t . Because u ∈ S(µ) it follows that Z

t+T

|y(τ )|2 dτ ≤

t

α12 [∆(t, t0 ) + c0 µT + c1 ] α02

(3.3.9)

4 Rt where ∆(t, t0 )) = t0 e−α0 (t−s) |u(s)|2 ds. If we establish that ∆(t, t0 ) ≤ c for some constant c independent of t, t0 then we can conclude from (3.3.9) that y ∈ S(µ). We start with Z t ∆(t, t0 ) = e−α0 (t−s) |u(s)|2 ds t0

3.3. INPUT/OUTPUT STABILITY ≤ ≤

e

−α0 t

e−α0 t

nt Z X i=0 nt X

85 i+1+t0

eα0 s |u(s)|2 ds

(3.3.10)

i+t0

Z eα0 (i+1+t0 )

i=0

i+1+t0

|u(s)|2 ds

i+t0

where nt is an integer that depends on t and satisfies nt + t0 ≤ t < nt + 1 + t0 . Because u ∈ S(µ), we have ∆(t, t0 ) ≤ e−α0 t (c0 µ + c1 )

nt X

eα0 (i+1+t0 ) ≤

i=0

c0 µ + c1 α0 e 1 − e−α0

(3.3.11)

Using (3.3.11) in (3.3.9) we have µ ¶ Z t+T c0 µ + c1 α0 α12 2 |y(τ )| dτ ≤ 2 c0 µT + c1 + e α0 1 − e−α0 t ³ ´ 2 c α2 c0 µ+c1 α0 α1 for any t ≥ t0 ≥ 0. Setting cˆ0 = 0α2 1 and cˆ1 = c1 + 1−e , it follows that −α0 e α20 0 y ∈ S(µ). From (3.3.6), (3.3.10), and (3.3.11), we can calculate the upper bound for |y(t)|2 .

2 Definition 3.3.1 may be generalized to the case where µ is not necessarily a constant as follows. Definition 3.3.2 Let x : [0, ∞) 7→ Rn , w : [0, ∞) 7→ R+ where x ∈ L2e , w ∈ L1e and consider the set ¯Z ) Z t+T ¯ t+T ¯ x, w ¯ x> (τ )x(τ )dτ ≤ c0 w(τ )dτ + c1 , ∀t, T ≥ 0 ¯ t t

(

S(w) =

where c0 , c1 ≥ 0 are some finite constants. We say that x is w-small in the m.s.s. if x ∈ S(w). We employ Corollary 3.3.3, and Definitions 3.3.1 and 3.3.2 repeatedly in Chapters 8 and 9 for the analysis of the robustness properties of adaptive control systems.

3.3.2

The L2δ Norm and I/O Stability

The definitions and results of the previous sections are very helpful in developing I/O stability results based on a different norm that are particularly useful in the analysis of adaptive schemes.

86

CHAPTER 3. STABILITY In this section we consider the exponentially weighted L2 norm defined

as 4

kxt k2δ =

µZ t 0

¶1

e

−δ(t−τ ) >

x (τ )x(τ )dτ

2

where δ ≥ 0 is a constant. We say that x ∈ L2δ if kxt k2δ exists. When δ = 0 we omit it from the subscript and use the notation x ∈ L2e . We refer to k(·)k2δ as the L2δ norm. For any finite time t, the L2δ norm satisfies the properties of the norm given by Definition 3.2.1, i.e., (i) kxt k2δ ≥ 0 (ii) kαxt k2δ = |α|kxt k2δ for any constant scalar α (iii) k(x + y)t k2δ ≤ kxt k2δ + kyt k2δ It also follows that (iv) kαxt k2δ ≤ kxt k2δ supt |α(t)| for any α ∈ L∞ The notion of L2δ norm has been introduced mainly to simplify the stability and robustness analysis of adaptive systems. To avoid confusion, we should point out that the L2δ norm defined here is different from the exponentially weighted norm used in many functional analysis books that is nR

o1

defined as 0t eδτ x> (τ )x(τ )dτ 2 . The main difference is that this exponentially weighted norm is a nondecreasing function of t, whereas the L2δ norm may not be. Let us consider the LTI system given by y = H(s)u

(3.3.12)

where H(s) is a rational function of s and examine L2δ stability, i.e., given u ∈ L2δ , what can we say about the Lp , L2δ properties of the output y(t) and its upper bounds. Lemma 3.3.2 Let H(s) in (3.3.12) be proper. If H(s) is analytic in Re[s] ≥ −δ/2 for some δ ≥ 0 and u ∈ L2e then (i) kyt k2δ ≤ kH(s)k∞δ kut k2δ where

¯ µ ¶¯ ¯ δ ¯¯ ¯ kH(s)k∞δ = sup ¯H jω − 2 ¯ ω 4

3.3. INPUT/OUTPUT STABILITY

87

(ii) Furthermore, when H(s) is strictly proper, we have |y(t)| ≤ kH(s)k2δ kut k2δ where kH(s)k2δ

1 =√ 2π 4

(Z )1 ¯ ¶¯2 2 ∞ ¯ µ ¯ δ ¯H jω − ¯ dω ¯ ¯ 2 −∞

The norms kH(s)k2δ , kH(s)k∞δ are related by the inequality 1 kH(s)k2δ ≤ √ k(s + p)H(s)k∞δ 2p − δ for any p > δ/2 ≥ 0. Proof The transfer function H(s) can be expressed as H(s) = d + Ha (s) with ½ 0 t<0 h(t) = dδ∆ (t) + ha (t) t ≥ 0 Because d is a finite constant, H(s) being analytic in Re[s] ≥ −δ/2 implies that ha ∈ L1 , i.e., the pair {H(s), h(t)} belongs to the class of functions considered in Remark 3.3.1. If we define ½ 0 t<0 hδ (t) = δ t 2 dδ∆ (t) + e ha (t) t ≥ 0 4

δ

4

δ

yδ (t) = e 2 t y(t) and uδ (t) = e 2 t u(t), it follows from (3.3.1) that Z t δ δ yδ (t) = e 2 (t−τ ) h(t − τ )e 2 τ u(τ )dτ = hδ ∗ uδ 0

Now u ∈ L2e ⇒ uδ ∈ L2e . Therefore, applying Lemma 3.3.1 and Remark 3.3.1 for the truncated signals yδt , uδt at time t and noting that H(s − δ/2) is the Laplace transform of hδ we have kyδt k2 ≤ kH(s − δ/2)k∞ kuδt k2 − δ2 t

− δ2 t

(3.3.13)

Because e kyδt k2 = kyt k2δ , e kuδt k2 = kut k2δ , and kH(s−δ/2)k∞ = kH(s)k∞δ , (i) follows directly from (3.3.13). For d = 0, i.e., H(s) is strictly proper, we have ¯Z t ¯ ¯ ¯ δ (t−τ ) − δ2 (t−τ ) ¯ 2 e |y(t)| ≤ ¯ h(t − τ )e u(τ )dτ ¯¯ 0

µZ ≤

t

δ(t−τ )

e 0

2

|h(t − τ )| dτ

¶12 kut k2δ

88

CHAPTER 3. STABILITY

where the second inequality is obtained by applying the Schwartz inequality. Then, µZ |y(t)|

∞

≤

e 0

=

1 √ 2π

δ(t−τ )

µZ

∞

2

¶21

|h(t − τ )| dτ

kut k2δ

|H(jω − δ/2)|2 dω

¶21 kut k2δ

(3.3.14)

−∞

where the equality is obtained by assuming that H(s) is strictly proper and applying Parseval’s Theorem [42](p. 236), implies (ii). because H(s) is strictly proper, we can write kH(s)k2δ

= ≤ =

µZ

¶21 1 |(jω + p0 )H(jω − δ/2)| dω |jω + p0 |2 −∞ µZ ∞ ¶ 21 1 1 √ dω sup(|(jω + p0 )H(jω − δ/2)|) 2 ω 2π −∞ |jω + p0 | 1 √ k(s + p0 + δ/2)H(s)k∞δ 2p0 1 √ 2π

∞

2

for any p0 > 0. Setting p0 = p − δ/2, the result follows.

2

Remark 3.3.3 Lemma 3.3.2 can be extended to the case where H(s) is not rational in s but belongs to the general class of transfer functions described in Remark 3.3.1. We refer to kH(s)k2δ , kH(s)k∞δ defined in Lemma 3.3.2 as the δ-shifted H2 and H∞ norms, respectively. Lemma 3.3.3 Consider the linear time-varying system given by x˙ = A(t)x + B(t)u, x(0) = x0 y = C > (t)x + D(t)u

(3.3.15)

where x ∈ Rn , y ∈ Rr , u ∈ Rm , and the elements of the matrices A, B, C, and D are bounded continuous functions of time. If the state transition matrix Φ(t, τ ) of (3.3.15) satisfies kΦ(t, τ )k ≤ λ0 e−α0 (t−τ )

(3.3.16)

for some λ0 , α0 > 0 and u ∈ L2e , then for any δ ∈ [0, δ1 ) where 0 < δ1 < 2α0 is arbitrary, we have

3.3. INPUT/OUTPUT STABILITY √ cλ0 kut k2δ + ²t 2α0 −δ cλ0 kxt k2δ ≤ √ kut k2δ (δ1 −δ)(2α0 −δ1 )

89

(i) |x(t)| ≤ (ii)

+ ²t

(iii) kyt k2δ ≤ c0 kut k2δ + ²t where cλ0 c0 = p sup kC > (t)k + sup kD(t)k, c = sup kB(t)k (δ1 − δ)(2α0 − δ1 ) t t t and ²t is an exponentially decaying to zero term because x0 6= 0. Proof The solution x(t) of (3.3.15) can be expressed as Z

t

x(t) = Φ(t, 0)x0 +

Φ(t, τ )B(τ )u(τ )dτ 0

Therefore,

Z

t

|x(t)| ≤ kΦ(t, 0)k|x0 | +

kΦ(t, τ )k kB(τ )k |u(τ )|dτ 0

Using (3.3.16) we have Z |x(t)| ≤ ²t + cλ0

t

e−α0 (t−τ ) |u(τ )|dτ

(3.3.17)

0

where c and λ0 are as defined in the statement of the lemma. Expressing e−α0 (t−τ ) δ δ as e−(α0 − 2 )(t−τ ) e− 2 (t−τ ) and applying the Schwartz inequality, we have µZ |x(t)| ≤

t

²t + cλ0

e

−(2α0 −δ)(t−τ )

0

≤

²t + √

¶12 µZ

t

dτ

e

−δ(t−τ )

2

¶21

|u(τ )| dτ

0

cλ0 kut k2δ 2α0 − δ

which completes the proof of (i). Using property (iii) of Definition 3.2.1 for the L2δ norm, it follows from (3.3.17) that kxt k2δ ≤ k²t k2δ + cλ0 kQt k2δ

(3.3.18)

where °µZ t ¶ −α0 (t−τ ) kQt k2δ= ° e |u(τ )|dτ ° 4°

0

° "Z t µZ τ ¶2 #21 ° −δ(t−τ ) −α0 (τ −s) ° = e e |u(s)|ds dτ °

t 2δ

0

0

90

CHAPTER 3. STABILITY

Using the Schwartz inequality we have µZ τ ¶2 µZ τ ¶2 δ1 δ1 e−α0 (τ −s) |u(s)|ds = e−(α0 − 2 )(τ −s) e− 2 (τ −s) |u(s)|ds 0 Z τ0 Z τ ≤ e−(2α0 −δ1 )(τ −s) ds e−δ1 (τ −s) |u(s)|2 ds 0 0 Z τ 1 ≤ e−δ1 (τ −s) |u(s)|2 ds 2α0 − δ1 0 i.e., kQt k2δ ≤ √

µZ

1 2α0 − δ1

Z

t

e

−δ(t−τ )

0

τ

e

−δ1 (τ −s)

2

¶12

|u(s)| dsdτ

(3.3.19)

0

Interchanging the sequence of integration, (3.3.19) becomes kQt k2δ

≤ = = ≤

√

1 2α0 − δ1

µZ

t

Z e−δt+δ1 s |u(s)|2

0

µZ

t

e−(δ1 −δ)τ dτ ds

¶12

s

¶12 − e−(δ1 −δ)t e |u(s)| ds δ1 − δ 2α0 − δ1 0 µZ t −δ(t−s) ¶21 1 e − e−δ1 (t−s) 2 √ |u(s)| ds δ1 − δ 2α0 − δ1 0 µZ t ¶21 1 −δ(t−s) 2 p e |u(s)| ds (2α0 − δ1 )(δ1 − δ) 0 √

1

t

2e

−δt+δ1 s

−(δ1 −δ)s

for any δ < δ1 < 2α0 . Because k²t k2δ ≤ ²t , the proof of (ii) follows. The proof of (iii) follows directly by noting that kyt k2δ ≤ k(C > x)t k2δ + k(Du)t k2δ ≤ kxt k2δ sup kC > (t)k + kut k2δ sup kD(t)k t

t

2 A useful extension of Lemma 3.3.3, applicable to the case where A(t) is not necessarily stable and δ = δ0 > 0 is a given fixed constant, is given by the following Lemma that makes use of the following definition. Definition 3.3.3 The pair (C(t),A(t)) in (3.3.15) is uniformly completely observable (UCO) if there exist constants β1 , β2 , ν > 0 such that for all t0 ≥ 0, β2 I ≥ N (t0 , t0 + ν) ≥ β1 I

3.3. INPUT/OUTPUT STABILITY 4

91

R

where N (t0 , t0 + ν) = tt00 +ν Φ> (τ, t0 )C(τ )C > (τ )Φ(τ, t0 )dτ is the so-called observability grammian [1, 201] and Φ(t, τ ) is the state transition matrix associated with A(t). Lemma 3.3.4 Consider a linear time-varying system of the same form as (3.3.15) where (C(t), A(t)) is UCO, and the elements of A, B, C, and D are bounded continuous functions of time. For any given finite constant δ0 > 0, we have (i) (ii)

|x(t)| ≤ √2αλ1−δ (c1 kut k2δ0 + c2 kyt k2δ0 ) + ²t 1 0 λ1 kx(t)k2δ0 ≤ √ (c1 kut k2δ0 + c2 kyt k2δ0 ) + ²1 (δ1 −δ0 )(2α1 −δ1 )

(iii) kyt k2δ0 ≤ kxt k2δ0 supt kC > (t)k + kut k2δ0 supt kD(t)k where c1 , c2 ≥ 0 are some finite constants; δ1 , α1 satisfy δ0 < δ1 < 2α1 , and ²t is an exponentially decaying to zero term because x0 6= 0. Proof Because (C, A) is uniformly completely observable, there exists a matrix ∆ K(t) with bounded elements such that the state transition matrix Φc (t, τ ) of Ac (t) = A(t) − K(t)C > (t) satisfies kΦc (t, τ )k ≤ λ1 e−α1 (t−τ ) for some constants α1 , δ1 , λ1 that satisfy α1 > δ21 > δ20 , λ1 > 0. Let us now rewrite (3.3.15), by using what is called “output injection,” as x˙ = (A − KC > )x + Bu + KC > x Because C > x = y − Du, we have ¯ + Ky x˙ = Ac (t)x + Bu ¯ = B − KD. Following exactly the same procedure as in the proof of where B Lemma 3.3.3, we obtain |x(t)| ≤ √

λ1 (c1 kut k2δ0 + c2 kyt k2δ0 ) + ²t 2α1 − δ0

¯ where c1 = supt kB(t)k, c2 = supt kK(t)k and ²t is an exponentially decaying to zero term due to x(0) = x0 . Similarly, λ1

kxt k2δ0 ≤ p

(δ1 − δ0 )(2α1 − δ1 )

(c1 kut k2δ0 + c2 kyt k2δ0 ) + ²t

92

CHAPTER 3. STABILITY

by following exactly the same steps as in the proof of Lemma 3.3.3. The proof of (iii) follows directly from the expression of y. 2

Instead of the interval [0, t), the L2δ norm can be defined over any arbitrary interval of time as follows: ∆

kxt,t1 k2δ =

µZ t t1

¶1

e

−δ(t−τ ) >

x (τ )x(τ )dτ

2

for any t1 ≥ 0 and t ≥ t1 . This definition allow us to use the properties of the L2δ norm over certain intervals of time that are of interest. We develop some of these properties for the LTI, SISO system x˙ = Ax + Bu,

x(0) = x0

>

y = C x + Du

(3.3.20)

whose transfer function is given by y = [C > (sI − A)−1 B + D]u = H(s)u

(3.3.21)

Lemma 3.3.5 Consider the LTI system (3.3.20), where A is a stable matrix and u ∈ L2e . Let α0 , λ0 be the positive constants that satisfy keA(t−τ ) k ≤ λ0 e−α0 (t−τ ) . Then for any constant δ ∈ [0, δ1 ) where 0 < δ1 < 2α0 is arbitrary, for any finite t1 ≥ 0 and t ≥ t1 we have (a) |x(t)| ≤ λ0 e−α0 (t−t1 ) |x(t1 )| + c1 kut,t1 k2δ (i) δ (b) kxt,t1 k2δ ≤ c0 e− 2 (t−t1 ) |x(t1 )| + c2 kut,t1 k2δ δ

(ii) kyt,t1 k2δ ≤ c3 e− 2 (t−t1 ) |x(t1 )| + kH(s)k∞δ kut,t1 k2δ (iii) Furthermore if D = 0, i.e., H(s) is strictly proper, then |y(t)| ≤ c4 e−α0 (t−t1 ) |x(t1 )| + kH(s)k2δ kut,t1 k2δ where λ0 c1 = kBkc0 , c0 = √ , 2α0 − δ c3 = kC > kc0 ,

kBkλ0 (δ1 − δ)(2α0 − δ1 )

c2 = p

c4 = kC > kλ0

3.3. INPUT/OUTPUT STABILITY Proof Define v(τ ) as

½ v(τ ) =

93

0 if τ < t1 u(τ ) if τ ≥ t1

From (3.3.20) we have x(t) = eA(t−t1 ) x(t1 ) + x ¯(t) where

Z

t

x ¯(t) =

eA(t−τ ) Bu(τ )dτ

∀t ≥ t1

(3.3.22)

∀t ≥ t1

t1

We can now rewrite x ¯(t) as Z

t

x ¯(t) =

eA(t−τ ) Bv(τ )dτ

∀t ≥ 0

(3.3.23)

0

Similarly

y(t) = C > eA(t−t1 ) x(t1 ) + y¯(t) ∀t ≥ t1 Z t y¯(t) = C > eA(t−τ ) Bv(τ )dτ + Dv(t) ∀t ≥ 0

(3.3.24) (3.3.25)

0

It is clear that x ¯ in (3.3.23) and y¯ in (3.3.25) are the solutions of the system x ¯˙ = y¯ =

A¯ x + Bv, x ¯(0) = 0 C >x ¯ + Dv

(3.3.26)

whose transfer function is C > (sI − A)−1 B + D = H(s). Because A is a stable matrix, there exists constants λ0 , α0 > 0 such that keA(t−τ ) k ≤ λ0 e−α0 (t−τ ) which also implies that H(s) is analytic in Re[s] ≥ −α0 . Let us now apply the results of Lemma 3.3.3 to (3.3.26). We have |¯ x(t)| ≤ k¯ xt k2δ

≤

kBkλ0 √ kvt k2δ = c1 kvt k2δ 2α0 − δ kBkλ0 p kvt k2δ = c2 kvt k2δ (δ1 − δ)(2α0 − δ1 )

for some δ1 > 0, δ > 0 such that 0 < δ < δ1 < 2α0 . Because kvt k2δ = kut,t1 k2δ and k¯ xt,t1 k2δ ≤ k¯ xt k2δ , it follows that for all t ≥ t1 |¯ x(t)| ≤ c1 kut,t1 k2δ ,

k¯ xt,t1 k2δ ≤ c2 kut,t1 k2δ

From (3.3.22) we have |x(t)| ≤ λ0 e−α0 (t−t1 ) |x(t1 )| + |¯ x(t)|

∀t ≥ t1

(3.3.27)

94

CHAPTER 3. STABILITY

which together with (3.3.27) imply (i)(a). Using (3.3.22) we have kxt,t1 k2δ ≤ k(eA(t−t1 ) x(t1 ))t,t1 k2δ + k¯ xt,t1 k2δ which implies that µZ kxt,t1 k2δ

t

−δ(t−τ ) −2α0 (τ −t1 )

e

≤

e

¶12 dτ

λ0 |x(t1 )| + k¯ xt,t1 k2δ

t1 δ

≤

λ0 e− 2 (t−t1 ) √ |x(t1 )| + k¯ xt,t1 k2δ 2α0 − δ

(3.3.28)

From (3.3.27) and (3.3.28), (i)(b) follows. Let us now apply the results of Lemma 3.3.2 to the system (3.3.26), also described by y¯ = H(s)v we have k¯ yt k2δ ≤ kH(s)k∞δ kvt k2δ and for H(s) strictly proper |¯ y (t)| ≤ kH(s)k2δ kvt k2δ for any 0 ≤ δ < 2α0 . Since kvt k2δ = kut,t1 k2δ and k¯ yt,t1 k2δ ≤ k¯ yt k2δ , we have k¯ yt,t1 k2δ ≤ kH(s)k∞δ kut,t1 k2δ

(3.3.29)

and |¯ y (t)| ≤ kH(s)k2δ kut,t1 k2δ ,

∀t ≥ t1

(3.3.30)

From (3.3.24) we have |y(t)| ≤ kC > kλ0 e−α0 (t−t1 ) |x(t1 )| + |¯ y (t)|,

∀t ≥ t1

(3.3.31)

which implies, after performing some calculations, that kyt,t1 k2δ ≤ kC > k √

δ λ0 e− 2 (t−t1 ) |x(t1 )| + k¯ yt,t1 k2δ , 2α0 − δ

∀t ≥ t1

(3.3.32)

Using (3.3.29) in (3.3.32) we establish (ii) and from (3.3.30) and (3.3.31) we establish (iii). 2

By taking t1 = 0, Lemma 3.3.5 also shows the effect of the initial condition x(0) = x0 of the system (3.3.20) on the bounds for |y(t)| and kyt k2δ . We can obtain a similar result as in Lemma 3.3.4 over the interval [t1 , t] by extending Lemma 3.3.5 to the case where A is not necessarily a stable matrix and δ = δ0 > 0 is a given fixed constant, provided (C, A) is an observable pair.

3.3. INPUT/OUTPUT STABILITY

95

Lemma 3.3.6 Consider the LTV system (3.3.15) where the elements of A(t), B(t), C(t), and D(t) are bounded continuous functions of time and whose state transition matrix Φ(t, τ ) satisfies kΦ(t, τ )k ≤ λ0 e−α0 (t−τ ) ∀t ≥ τ and t, τ ∈ [t1 , t2 ) for some t2 > t1 ≥ 0 and α0 , λ0 > 0. Then for any δ ∈ [0, δ1 ) where 0 < δ1 < 2α0 is arbitrary, we have √ cλ0 kut,t k2δ 1 2α0 −δ cλ0 − 2δ (t−t1 ) √ λ0 √ e |x(t )| + kut,t1 k2δ , ∀t 1 2α0 −δ (δ1 −δ)(2α0 −δ1 )

(i)

|x(t)| ≤ λ0 e−α0 (t−t1 ) |x(t1 )| +

(ii)

kxt,t1 k2δ ≤

∈ [t1 , t2 )

where c = supt kB(t)k. Proof The solution x(t) of (3.3.15) is given by Z t x(t) = Φ(t, t1 )x(t1 ) + Φ(t, τ )B(τ )u(τ )dτ t1

Hence,

Z

t

|x(t)| ≤ λ0 e−α0 (t−t1 ) |x(t1 )| + cλ0

e−α0 (t−τ ) |u(τ )|dτ

t1

Proceeding as in the proof of Lemma 3.3.3 we establish (i). Now µZ kxt,t1 k2δ kQt,t1 k2δ

t

−δ(t−τ ) −2α0 (τ −t1 )

≤ λ0 |x(t1 )| e e t1 °µZ ¶ ° t 4 ° −α0 (t−τ ) = ° e |u(τ )|dτ ° t1 t,t

1

¶12 dτ

+ cλ0 kQt,t1 k2δ

° ° ° ° °

2δ

Following exactly the same step as in the proof of Lemma 3.3.3 we establish that 1

kQt,t1 k2δ ≤ p

(2α0 − δ1 )(δ1 − δ)

Because

kut,t1 k2δ

λ0 |x(t1 )| − δ (t−t1 ) kxt,t1 k2δ ≤ √ e 2 + kQt,t1 k2δ 2α0 − δ

2

the proof of (ii) follows. Example 3.3.3 (i) Consider the system described by y = H(s)u

96

CHAPTER 3. STABILITY

where H(s) =

2 s+3 .

We have

kH(s)k∞δ

¯ ¯ 2 ¯ = sup ¯ ω ¯ jω + 3 −

¯ ¯ 4 ¯ , = δ ¯¯ 6 − δ 2

∀δ ∈ [0, 6)

and kH(s)k2δ

1 =√ 2π

ÃZ

∞

! 12

4

−∞ ω 2

+

(6−δ)2 4

For u(t) = 1, ∀t ≥ 0, we have y(t) = inequality (ii) of Lemma 3.3.2, i.e.,

2 3 (1

=√

dω

2 , 6−δ

∀δ ∈ [0, 6)

− e−3t ), which we can use to verify

2 2 |y(t)| = |1 − e−3t | ≤ √ 3 6−δ

µ

1 − e−δt δ

¶12

holds ∀t ∈ [0, ∞) and δ ∈ (0, 6). (ii) The system in (i) may also be expressed as y˙ = −3y + 2u,

y(0) = 0

Its transition matrix Φ(t, 0) = e−3t and from Lemma 3.3.3, we have |y(t)| ≤ √

2 kut k2δ , 6−δ

∀δ ∈ [0, 6)

For u(t) = 1, ∀t ≥ 0, the above inequality implies 2 2 |y(t)| = |1 − e−3t | ≤ √ 3 6−δ which holds for all δ ∈ (0, 6).

3.3.3

µ

1 − e−δt δ

¶12

5

Small Gain Theorem

Many feedback systems, including adaptive control systems, can be put in the form shown in Figure 3.1. The operators H1 , H2 act on e1 , e2 to produce the outputs y1 , y2 ; u1 , u2 are external inputs. Sufficient conditions for H1 , H2 to guarantee existence and uniqueness of solutions e1 , y1 , e2 , y2 for given inputs u1 , u2 in Lpe are discussed in [42]. Here we assume that H1 , H2 are such that the existence and uniqueness of solutions are guaranteed. The

3.3. INPUT/OUTPUT STABILITY

u1 + e1 - Σl − 6

97

y1

H1

y2

H2

¾

+ e2 ? ¾ u2 Σl +

Figure 3.1 Feedback system. problem is to determine conditions on H1 , H2 so that if u1 , u2 are bounded in some sense, then e1 , e2 , y1 , y2 are also bounded in the same sense. Let L be a normed linear space defined by 4

©

ª

L = f : R+ 7→ Rn | kf k < ∞

where k · k corresponds to any of the norms introduced earlier. Let Le be the extended normed space associated with L, i.e., ¯

©

Le = f : R+ 7→ Rn ¯kft k < ∞, ∀t ∈ R+ where

(

ft (τ ) =

ª

f (τ ) τ ≤ t 0 τ >t

The following theorem known as the small gain theorem [42] gives sufficient conditions under which bounded inputs produce bounded outputs in the feedback system of Figure 3.1. Theorem 3.3.3 Consider the system shown in Figure 3.1. Suppose H1 , H2 : Le 7→ Le ; e1 , e2 ∈ Le . Suppose that for some constants γ1 , γ2 ≥ 0 and β1 , β2 , the operators H1 , H2 satisfy k(H1 e1 )t k ≤ γ1 ke1t k + β1 k(H2 e2 )t k ≤ γ2 ke2t k + β2 ∀t ∈ R+ . If γ1 γ2 < 1 then

98 (i)

CHAPTER 3. STABILITY ke1t k ≤ (1 − γ1 γ2 )−1 (ku1t k + γ2 ku2t k + β2 + γ2 β1 ) ke2t k ≤ (1 − γ1 γ2 )−1 (ku2t k + γ1 ku1t k + β1 + γ1 β2 ) (3.3.33) for any t ≥ 0.

(ii) If in addition, ku1 k, ku2 k < ∞, then e1 , e2 , y1 , y2 have finite norms, and the norms of e1 , e2 are bounded by the right-hand sides of (3.3.33) with all subscripts t dropped. The constants γ1 , γ2 are referred to as the gains of H1 , H2 respectively. When u2 ≡ 0, there is no need to separate the gain of H1 and H2 . In this case, one can consider the “loop gain ” H2 H1 as illustrated by the following corollary: Corollary 3.3.4 Consider the system of Figure 3.1 with u2 ≡ 0. Suppose that k(H2 H1 e1 )t k ≤ γ21 ke1t k + β21 k(H1 e1 )t k ≤ γ1 ke1t k + β1 ∀t ∈ R+ for some constants γ21 , γ1 ≥ 0 and β21 , β1 . If γ21 < 1, then (i) ke1t k ≤ (1 − γ21 )−1 (ku1t k + β21 ) ky1t k ≤ γ1 (1 − γ21 )−1 (ku1t k + β21 ) + β1

(3.3.34)

for any t ≥ 0. (ii) If in addition ku1 k < ∞, then e1 , e2 , y1 , y2 have finite norms and (3.3.34) holds without the subscript t. The proofs of Theorem 3.3.3 and Corollary 3.3.4 follow by using the properties of the norm [42]. The small gain theorem is a very general theorem that applies to both continuous and discrete-time systems with multiple inputs and outputs. As we mentioned earlier, Theorem 3.3.3 and Corollary 3.3.4 assume the existence of solutions e1 , e2 ∈ Le . In practice, u1 , u2 are given external inputs and e1 , e2 are calculated using the operators H1 , H2 . Therefore, the existence of e1 , e2 ∈ Le depends on the properties of H1 , H2 . Example 3.3.4 Let us consider the feedback system

3.3. INPUT/OUTPUT STABILITY r + e1 - Σl − 6

99 y

- G(s)

K

-

+ e2 ? ¾ d Σl +

¾

−αs

where G(s) = es+2 , α ≥ 0 is a constant time delay, and K is a constant feedback gain. The external input r is an input command, and d is a noise disturbance. We are interested in finding conditions on the gain K such that (i) (ii)

r, d ∈ L∞ =⇒ e1 , e2 , y ∈ L∞ r, d ∈ L2 =⇒ e1 , e2 , y ∈ L2

The system is in the form of the general feedback system given in Figure 3.1, i.e., u1 = r, u2 = d Z t−α H1 e1 (t) = e2α e−2(t−τ ) e1 (τ )dτ 0

H2 e2 (t) = Ke2 (t) where e−2(t−α) for t ≥ α comes from the impulse response g(t) of G(s), i.e., g(t) = e−2(t−α) for t ≥ α and g(t) = 0 for t < α. (i) Because Z t−α |H1 e1 (t)| ≤ e2α e−2(t−τ ) |e1 (τ )|dτ 0 Z t−α ≤ e2α e−2(t−τ ) dτ ke1t k∞ 0

≤

1 ke1t k∞ 2

we have γ1 = 12 . Similarly, the L∞ -gain of H2 is γ2 = |K|. Therefore, for L∞ stability the small gain theorem requires |K| < 1, 2

i.e.,

|K| < 2

(ii) From Lemma 3.3.1, we have ¯ −αjω ¯ ¯e ¯ ¯ ke1t k2 = 1 ke1t k2 k(H1 e1 )t k2 ≤ sup ¯¯ 2 + jω ¯ 2 ω which implies that the L2 gain of H1 is γ1 = 12 . Similarly the L2 gain of H2 is γ2 = |K|, and the condition for L2 -stability is |K| < 2.

100

CHAPTER 3. STABILITY

For this simple system, however, with α = 0, we can verify that r, d ∈ L∞ =⇒ e1 , e2 , y ∈ L∞ if and only if K > −2, which indicates that the condition given by the small gain theorem (for α = 0) is conservative. 5 Example 3.3.5 Consider the system x˙ = Ac x,

Ac = A + B

where x ∈ Rn , A is a stable matrix, i.e., all the eigenvalues of A are in Re[s] < 0 and B is a constant matrix. We are interested in obtaining an upper bound for B such that Ac is a stable matrix. Let us represent the system in the form of Figure 3.1 as the following: H1 u1 = 0- l e1-+ l Σ + Σ+ + 6 6

-

1 I s

x

A

¾

B

¾

e2

-

+ ?u = 0 ¾2 Σl +

H2 1 We can verify that the L∞ gain of H1 is γ1 = α α0 where α1 , α0 > 0 are the constants in the bound keA(t−τ ) k ≤ α1 e−α0 (t−τ ) that follows from the stability of A. The L∞ gain of H2 is γ2 = kBk. Therefore for L∞ stability, we should have α1 kBk <1 α0 or α0 kBk < α1 Now L∞ stability implies that Ac = A + B is a stable matrix. (Note that the initial condition for x is taken to be zero, i.e., x(0) = 0.) 5

Despite its conservatism, the small gain theorem is widely used to design robust controllers for uncertain systems. In many applications, certain loop transformations are needed to transform a given feedback system to the form of the feedback system of Figure 3.1 where H1 , H2 have finite gains [42].

3.3. INPUT/OUTPUT STABILITY

3.3.4

101

Bellman-Gronwall Lemma

A key lemma for analysis of adaptive control schemes is the following. Lemma 3.3.7 (Bellman-Gronwall Lemma I) [232] Let λ(t), g(t), k(t) be nonnegative piecewise continuous functions of time t. If the function y(t) satisfies the inequality y(t) ≤ λ(t) + g(t) then y(t) ≤ λ(t) + g(t)

Z t t0

Z t

k(s)y(s)ds,

t0

·

λ(s)k(s) exp

µZ t

∀t ≥ t0 ≥ 0

(3.3.35)

¶¸

k(τ )g(τ )dτ

s

ds

∀t ≥ t0 ≥ 0 (3.3.36)

In particular, if λ(t) ≡ λ is a constant and g(t) ≡ 1, then µZ t

y(t) ≤ λexp

t0

¶

k(s)ds

Proof Let us define 4

q(t) = k(t)e

−

Rt

∀t ≥ t0 ≥ 0

g(τ )k(τ )dτ

t0

Because k(t) is nonnegative, we have q(t) ≥ 0 ∀t ≥ t0 . Multiplying both sides of (3.3.35) by q(t), and rearranging the inequality we obtain Z t q(t)y(t) − q(t)g(t) k(s)y(s)ds ≤ λ(t)q(t) (3.3.37) t0

From the expression of q(t), one can verify that µ Rt ¶ Z t Z t − g(τ )k(τ )dτ d q(t)y(t) − q(t)g(t) k(s)y(s)ds = e t0 k(s)y(s)ds (3.3.38) dt t0 t0 Using (3.3.38) in (3.3.37) and integrating both sides of (3.3.37), we obtain Rt Z t Z t − g(τ )k(τ )dτ t0 e k(s)y(s)ds ≤ λ(s)q(s)ds t0

Therefore, Z

Rt

t

k(s)y(s)ds

≤ e

t0

t0

Rt = e Z

t0

g(τ )k(τ )dτ

g(τ )k(τ )dτ

t0

Z

t

λ(s)q(s)ds t0 Z t

λ(s)k(s)e

−

Rs t0

g(τ )k(τ )dτ

ds

t0

Rt

t

=

λ(s)k(s)e t0

s

g(τ )k(τ )dτ

ds

(3.3.39)

102

CHAPTER 3. STABILITY

Using (3.3.39) in (3.3.35), the proof for the inequality (3.3.36) is complete. Consider the special case where λ is a constant and g = 1. Define Z

4

t

k(s)y(s)ds

q1 = λ + t0

From (3.3.35), we have y(t) ≤ q1 (t) Now q˙1 = ky Because k ≥ 0, we have q˙1 ≤ kq1 Let w = q˙1 − kq1 . Clearly, w ≤ 0 and q˙1 = kq1 + w which implies

Rt q1 (t) = e

t0

k(τ )dτ

Z

Rt

t

q1 (t0 ) +

e

τ

k(s)ds

w(τ )dτ

(3.3.40)

t0

Because k ≥ 0, w ≤ 0 ∀t ≥ t0 and q1 (t0 ) = λ, it follows from (3.3.40) that Rt k(τ )dτ y(t) ≤ q1 (t) ≤ λe t0 and the proof is complete.

2

The reader can refer to [32, 232] for alternative proofs of the B-G Lemma. Other useful forms of the B-G lemma are given by Lemmas 3.3.8 and 3.3.9. Lemma 3.3.8 (B-G Lemma II) Let λ(t), k(t) be nonnegative piecewise continuous function of time t and let λ(t) be differentiable. If the function y(t) satisfies the inequality y(t) ≤ λ(t) + then

Rt

y(t) ≤ λ(t0 )e

t0

k(s)ds

Z t t0

+

k(s)y(s)ds, ∀t ≥ t0 ≥ 0

Z t t0

Rt

˙ λ(s)e

s

k(τ )dτ

ds, ∀t ≥ t0 ≥ 0.

3.3. INPUT/OUTPUT STABILITY Proof Let

103 Z

t

z(t) = λ(t) +

k(s)y(s)ds t0

it follows that z is differentiable and z ≥ y. We have z˙ = λ˙ + ky, z(t0 ) = λ(t0 ) Let v = z − y, then

z˙ = λ˙ + kz − kv

whose state transition matrix is Z

t

Φ(t, τ ) = exp

k(s)ds τ

Therefore,

Z

t

z(t) = Φ(t, t0 )z(t0 ) +

˙ ) − k(τ )v(τ )]dτ Φ(t, τ )[λ(τ

t0

Because

Z

t

Φ(t, τ )k(τ )v(τ )dτ ≥ 0 t0

resulting from Φ(t, τ ), k(τ ), v(τ ) being nonnegative, we have Z t ˙ )dτ z(t) ≤ Φ(t, t0 )z(t0 ) + Φ(t, τ )λ(τ t0

Using the expression for Φ(t, t0 ) in the above inequality, we have Rt Z t Rt k(s)ds k(τ )dτ ˙ s y(t) ≤ z(t) ≤ λ(t0 )e t0 + λ(s)e ds t0

2

and the proof is complete.

Lemma 3.3.9 (B-G Lemma III) Let c0 , c1 , c2 , α be nonnegative constants and k(t) a nonnegative piecewise continuous function of time. If y(t) satisfies the inequality y(t) ≤ c0 e−α(t−t0 ) + c1 + c2

Z t t0

e−α(t−τ ) k(τ )y(τ )dτ, ∀t ≥ t0

then y(t) ≤ (c0 + c1 )e

−α(t−t0 ) c2

e

Rt t0

k(s)ds

+ c1 α

Z t t0

−α(t−τ ) c2

e

e

Rt τ

k(s)ds

dτ, ∀t ≥ t0

104

CHAPTER 3. STABILITY

Proof The proof follows directly from Lemma 3.3.8 by rewriting the given inequality of y as Z t

y¯(t) ≤ λ(t) +

¯ )¯ k(τ y (τ )dτ

t0

¯ where y¯(t) = eαt y(t), k(t) = c2 k(t), λ(t) = c0 eαt0 + c1 eαt . Applying Lemma 3.3.8, we obtain Rt Z t Rt c k(s)ds αt αt0 c2 t0 k(s)ds e y(t) ≤ (c0 + c1 )e e + c1 α eατ e 2 τ dτ t0

The result follows by multiplying each side of the above inequality by e−αt .

2

The B-G Lemma allows us to obtain an explicit bound for y(t) from the implicit bound of y(t) given by the integral inequality (3.3.35). Notice that if y(t) ≥ 0 and λ(t) = 0 ∀t ≥ 0, (3.3.36) implies that y(t) ≡ 0 ∀t ≥ 0. In many cases, the B-G Lemma may be used in place of the small gain theorem to analyze a class of feedback systems in the form of Figure 3.1 as illustrated by the following example. Example 3.3.6 Consider the same system as in Example 3.3.5. We have Z t x(t) = eAt x(0) + eA(t−τ ) Bx(τ )dτ 0

Hence,

Z

t

|x(t)| ≤ α1 e−α0 t |x(0)| +

α1 e−α0 (t−τ ) kBk|x(τ )|dτ

0

i.e.,

Z |x(t)| ≤ α1 e

−α0 t

−α0 t

|x(0)| + α1 e

kBk

t

eα0 τ |x(τ )|dτ

0

Applying the B-G Lemma I with λ = α1 e−α0 t |x(0)|, g(t) = α1 kBke−α0 t , k(t) = eα0 t , we have |x(t)| ≤ α1 e−α0 t |x(0)| + α1 |x(0)|e−γt where γ = α0 − α1 kBk Therefore, for |x(t)| to be bounded from above by a decaying exponential (which implies that Ac = A + B is a stable matrix), B has to satisfy kBk <

α0 α1

3.4. LYAPUNOV STABILITY

105

which is the same condition we obtained in Example 3.3.5 using the small gain theorem. In this case, we assume that |x(0)| 6= 0, otherwise for x(0) = 0 we would have λ(t) = 0 and |x(t)| = 0 ∀t ≥ 0 which tells us nothing about the stability of Ac . The reader may like to verify the same result using B-G Lemmas II and III. 5

3.4

Lyapunov Stability

3.4.1

Definition of Stability

We consider systems described by ordinary differential equations of the form x˙ = f (t, x),

x(t0 ) = x0

(3.4.1)

where x ∈ Rn , f : J ×B(r) 7→ R, J = [t0 , ∞) and B(r) = {x ∈ Rn | |x| < r}. We assume that f is of such nature that for every x0 ∈ B(r) and every t0 ∈ R+ , (3.4.1) possesses one and only one solution x(t; t0 , x0 ). Definition 3.4.1 A state xe is said to be an equilibrium state of the system described by (3.4.1) if f (t, xe ) ≡ 0

f or all t ≥ t0

Definition 3.4.2 An equilibrium state xe is called an isolated equilib4 rium state if there exists a constant r > 0 such that B(xe , r) = {x | |x − xe | < r} ⊂ Rn contains no equilibrium state of (3.4.1) other than xe . The equilibrium state x1e = 0, x2e = 0 of x˙ 1 = x1 x2 , x˙ 2 = x21 is not isolated because any point x1 = 0, x2 = constant is an equilibrium state. The differential equation x˙ = (x − 1)2 x has two isolated equilibrium states xe = 1 and xe = 0.

106

CHAPTER 3. STABILITY

Definition 3.4.3 The equilibrium state xe is said to be stable (in the sense of Lyapunov) if for arbitrary t0 and ² > 0 there exists a δ(², t0 ) such that |x0 − xe | < δ implies |x(t; t0 , x0 ) − xe | < ² for all t ≥ t0 . Definition 3.4.4 The equilibrium state xe is said to be uniformly stable (u.s.) if it is stable and if δ(², t0 ) in Definition 3.4.3 does not depend on t0 . Definition 3.4.5 The equilibrium state xe is said to be asymptotically stable (a.s.) if (i) it is stable, and (ii) there exists a δ(t0 ) such that |x0 − xe | < δ(t0 ) implies limt→∞ |x(t; t0 , x0 ) − xe | = 0. Definition 3.4.6 The set of all x0 ∈ Rn such that x(t; t0 , x0 ) → xe as t → ∞ for some t0 ≥ 0 is called the region of attraction of the equilibrium state xe . If condition (ii) of Definition 3.4.5 is satisfied, then the equilibrium state xe is said to be attractive. Definition 3.4.7 The equilibrium state xe is said to be uniformly asymptotically stable (u.a.s.) if (i) it is uniformly stable, (ii) for every ² > 0 and any t0 ∈ R+ , there exist a δ0 > 0 independent of t0 and ² and a T (²) > 0 independent of t0 such that |x(t; t0 , x0 ) − xe | < ² for all t ≥ t0 + T (²) whenever |x0 − xe | < δ0 . Definition 3.4.8 The equilibrium state xe is exponentially stable (e.s.) if there exists an α > 0 , and for every ² > 0 there exists a δ(²) > 0 such that |x(t; t0 , x0 ) − xe | ≤ ²e−α(t−t0 ) f or all t ≥ t0 whenever |x0 − xe | < δ(²). Definition 3.4.9 The equilibrium state xe is said to be unstable if it is not stable. When (3.4.1) possesses a unique solution for each x0 ∈ Rn and t0 ∈ R+ , we need the following definitions for the global characterization of solutions. Definition 3.4.10 A solution x(t; t0 , x0 ) of (3.4.1) is bounded if there exists a β > 0 such that |x(t; t0 , x0 )| < β for all t ≥ t0 , where β may depend on each solution.

3.4. LYAPUNOV STABILITY

107

Definition 3.4.11 The solutions of (3.4.1) are uniformly bounded (u.b.) if for any α > 0 and t0 ∈ R+ , there exists a β = β(α) independent of t0 such that if |x0 | < α, then |x(t; t0 , x0 )| < β for all t ≥ t0 . Definition 3.4.12 The solutions of (3.4.1) are uniformly ultimately bounded (u.u.b.) (with bound B) if there exists a B > 0 and if corresponding to any α > 0 and t0 ∈ R+ , there exists a T = T (α) > 0 (independent of t0 ) such that |x0 | < α implies |x(t; t0 , x0 )| < B for all t ≥ t0 + T . Definition 3.4.13 The equilibrium point xe of (3.4.1) is asymptotically stable in the large (a.s. in the large) if it is stable and every solution of (3.4.1) tends to xe as t → ∞ (i.e., the region of attraction of xe is all of Rn ). Definition 3.4.14 The equilibrium point xe of (3.4.1) is uniformly asymptotically stable in the large (u.a.s. in the large) if (i) it is uniformly stable, (ii) the solutions of (3.4.1) are uniformly bounded, and (iii) for any α > 0, any ² > 0 and t0 ∈ R+ , there exists T (², α) > 0 independent of t0 such that if |x0 − xe | < α then |x(t; t0 , x0 ) − xe | < ² for all t ≥ t0 + T (², α). Definition 3.4.15 The equilibrium point xe of (3.4.1) is exponentially stable in the large (e.s. in the large) if there exists α > 0 and for any β > 0, there exists k(β) > 0 such that |x(t; t0 , x0 )| ≤ k(β)e−α(t−t0 )

f or all t ≥ t0

whenever |x0 | < β. Definition 3.4.16 If x(t; t0 , x0 ) is a solution of x˙ = f (t, x), then the trajectory x(t; t0 , x0 ) is said to be stable (u.s., a.s., u.a.s., e.s., unstable) if the equilibrium point ze = 0 of the differential equation z˙ = f (t, z + x(t; t0 , x0 )) − f (t, x(t; t0 , x0 )) is stable (u.s., a.s., u.a.s., e.s., unstable, respectively). The above stability concepts and definitions are illustrated by the following example:

108

CHAPTER 3. STABILITY

Example 3.4.1 (i)

x˙ = 0 has the equilibrium state xe = c, where c is any constant, which is not an isolated equilibrium state. It can be easily verified that xe = c is stable, u.s. but not a.s. (ii) x˙ = −x3 has an isolated equilibrium state xe = 0. Its solution is given by µ x(t) = x(t; t0 , x0 ) =

x20 1 + 2x20 (t − t0 )

¶21

Now given any ² > 0, |x0 | < δ = ² implies that s x20 |x(t)| = ≤ |x0 | < ² ∀t ≥ t0 ≥ 0 1 + 2x20 (t − t0 )

(3.4.2)

Hence, according to Definition 3.4.3, xe = 0 is stable. Because δ = ² is independent of t0 , xe = 0 is also u.s. Furthermore, because xe = 0 is stable and x(t) → xe = 0 as t → ∞ for all x0 ∈ R, we have a.s. in the large. Let us now check whether xe = 0 is u.a.s. in the large by using Definition 3.4.14. We have already shown u.s. From (3.4.2) we have that x(t) is u.b. To satisfy condition (iii) of Definition 3.4.14, we need to find a T > 0 independent of t0 such that for any α > 0 and ² > 0, |x0 | < α implies |x(t)| < ² for all t ≥ t0 + T . From (3.4.2) we have s r x20 1 |x(t)| ≤ |x(t0 + T )| = < , ∀t ≥ t0 + T 1 + 2x20 T 2T Choosing T = 2²12 , it follows that |x(t)| < ² ∀t ≥ t0 + T . Hence, xe = 0 is u.a.s. in the large. Using Definition 3.4.15, we can conclude that xe = 0 is not e.s. (iii) x˙ = (x − 2)x has two isolated equilibrium states xe = 0 and xe = 2. It can be shown that xe = 0 is e.s. with the region of attraction Ra = {x | x < 2} and xe = 2 is unstable. 1 x has an equilibrium state xe = 0 that is stable, u.s., a.s. in the large (iv) x˙ = − 1+t but is not u.a.s. (verify). (v) x˙ = (t sin t − cos t − 2)x has an isolated equilibrium state xe = 0 that is stable, a.s. in the large but not u.s. (verify). ∇

3.4.2

Lyapunov’s Direct Method

The stability properties of the equilibrium state or solution of (3.4.1) can be studied by using the so-called direct method of Lyapunov (also known as Lyapunov’s second method) [124, 125]. The objective of this method is

3.4. LYAPUNOV STABILITY

109

to answer questions of stability by using the form of f (t, x) in (3.4.1) rather than the explicit knowledge of the solutions. We start with the following definitions [143]. Definition 3.4.17 A continuous function ϕ : [0, r] 7→ R+ (or a continuous function ϕ : [0, ∞) 7→ R+ ) is said to belong to class K, i.e., ϕ ∈ K if (i) (ii)

ϕ(0) = 0 ϕ is strictly increasing on [0, r] (or on [0, ∞)).

Definition 3.4.18 A continuous function ϕ : [0, ∞) 7→ R+ is said to belong to class KR, i.e., ϕ ∈ KR if (i) ϕ(0) = 0 (ii) ϕ is strictly increasing on [0, ∞) (iii) limr→∞ ϕ(r) = ∞. 2

x The function ϕ(|x|) = 1+x 2 belongs to class K defined on [0, ∞) but not to class KR. The function ϕ(|x|) = |x| belongs to class K and class KR. It is clear that ϕ ∈ KR implies ϕ ∈ K, but not the other way.

Definition 3.4.19 Two functions ϕ1 , ϕ2 ∈ K defined on [0, r] (or on [0, ∞)) are said to be of the same order of magnitude if there exist positive constants k1 .k2 such that k1 ϕ1 (r1 ) ≤ ϕ2 (r1 ) ≤ k2 ϕ1 (r1 ), ∀r1 ∈ [0, r] ( or ∀r1 ∈ [0, ∞)) The function ϕ1 (|x|) = magnitude (verify!).

x2 1+2x2

and ϕ2 =

x2 1+x2

are of the same order of

Definition 3.4.20 A function V (t, x) : R+ × B(r) 7→ R with V (t, 0) = 0 ∀t ∈ R+ is positive definite if there exists a continuous function ϕ ∈ K such that V (t, x) ≥ ϕ(|x|) ∀t ∈ R+ , x ∈ B(r) and some r > 0. V (t, x) is called negative definite if −V (t, x) is positive definite. The function V (t, x) = V (t, x) = all x ∈ R.

1 2 1+t x

x2 1−x2

with x ∈ B(1) is positive definite, whereas

is not. The function V (t, x) =

x2 1+x2

is positive definite for

110

CHAPTER 3. STABILITY

Definition 3.4.21 A function V (t, x) : R+ × B(r) 7→ R with V (t, 0) = 0 ∀t ∈ R+ is said to be positive (negative) semidefinite if V (t, x) ≥ 0 (V (t, x) ≤ 0) for all t ∈ R+ and x ∈ B(r) for some r > 0. Definition 3.4.22 A function V (t, x) : R+ × B(r) 7→ R with V (t, 0) = 0 ∀t ∈ R+ is said to be decrescent if there exists ϕ ∈ K such that |V (t, x)| ≤ ϕ(|x|) ∀t ≥ 0 and ∀x ∈ B(r) for some r > 0. 1 2 The function V (t, x) = 1 + t x is decrescent because V (t, x) = 2 + 2 x ∀t ∈ R but V (t, x) = tx is not.

1 2 1 + tx

≤

Definition 3.4.23 A function V (t, x) : R+ ×Rn 7→ R with V (t, 0) = 0 ∀t ∈ R+ is said to be radially unbounded if there exists ϕ ∈ KR such that V (t, x) ≥ ϕ(|x|) for all x ∈ Rn and t ∈ R+ . x2 satisfies conditions (i) and (ii) of Definition 3.4.23 1+x2 |x|2 (i.e., choose ϕ(|x|) = 1+|x|2 ). However, because V (x) ≤ 1, one cannot find a function ϕ(|x|) ∈ KR to satisfy V (x) ≥ ϕ(|x|) for all x ∈ Rn . Hence, V is

The function V (x) =

not radially unbounded. It is clear from Definition 3.4.23 that if V (t, x) is radially unbounded, it is also positive definite for all x ∈ Rn but the converse is not true. The reader should be aware that in some textbooks “positive definite” is used for radially unbounded functions, and “locally positive definite” is used for our definition of positive definite functions. Let us assume (without loss of generality) that xe = 0 is an equilibrium point of (3.4.1) and define V˙ to be the time derivative of the function V (t, x) along the solution of (3.4.1), i.e., ∂V V˙ = + (∇V )> f (t, x) ∂t

(3.4.3)

∂V ∂V ∂V > where ∇V = [ ∂x , , . . . , ∂x ] is the gradient of V with respect to x. The n 1 ∂x2 second method of Lyapunov is summarized by the following theorem.

Theorem 3.4.1 Suppose there exists a positive definite function V (t, x) : R+ ×B(r) 7→ R for some r > 0 with continuous first-order partial derivatives with respect to x, t, and V (t, 0) = 0 ∀t ∈ R+ . Then the following statements are true:

3.4. LYAPUNOV STABILITY (i) (ii) (iii) (iv)

111

If V˙ ≤ 0, then xe = 0 is stable. If V is decrescent and V˙ ≤ 0, then xe = 0 is u.s. If V is decrescent and V˙ < 0, then xe is u.a.s. If V is decrescent and there exist ϕ1 , ϕ2 , ϕ3 ∈ K of the same order of magnitude such that ϕ1 (|x|) ≤ V (t, x) ≤ ϕ2 (|x|), V˙ (t, x) ≤ −ϕ3 (|x|) for all x ∈ B(r) and t ∈ R+ , then xe = 0 is e.s.

In the above theorem, the state x is restricted to be inside the ball B(r) for some r > 0. Therefore, the results (i) to (iv) of Theorem 3.4.1 are referred to as local results. Statement (iii) is equivalent to that there exist ϕ1 , ϕ2 , ϕ3 ∈ K, where ϕ1 , ϕ2 , ϕ3 do not have to be of the same order of magnitude, such that ϕ1 (|x|) ≤ V (t, x) ≤ ϕ2 (|x|), V˙ (t, x) ≤ −ϕ3 (|x|). Theorem 3.4.2 Assume that (3.4.1) possesses unique solutions for all x0 ∈ Rn . Suppose there exists a positive definite, decrescent and radially unbounded function V (t, x) : R+ ×Rn 7→ R+ with continuous first-order partial derivatives with respect to t, x and V (t, 0) = 0 ∀t ∈ R+ . Then the following statements are true: (i) If V˙ < 0, then xe = 0 is u.a.s. in the large. (ii) If there exist ϕ1 , ϕ2 , ϕ3 ∈ KR of the same order of magnitude such that ϕ1 (|x|) ≤ V (t, x) ≤ ϕ2 (|x|), V˙ (t, x) ≤ −ϕ3 (|x|) then xe = 0 is e.s. in the large. Statement (i) of Theorem 3.4.2 is also equivalent to that there exist ϕ1 , ϕ2 ∈ K and ϕ3 ∈ KR such that ϕ1 (|x|) ≤ V (t, x) ≤ ϕ2 (|x|), V˙ (t, x) ≤ −ϕ3 (|x|), ∀x ∈ Rn For a proof of Theorem 3.4.1, 3.4.2, the reader is referred to [32, 78, 79, 97, 124]. Theorem 3.4.3 Assume that (3.4.1) possesses unique solutions for all x0 ∈ Rn . If there exists a function V (t, x) defined on |x| ≥ R (where R may be large) and t ∈ [0, ∞) with continuous first-order partial derivatives with respect to x, t and if there exist ϕ1 , ϕ2 ∈ KR such that

112 (i) (ii)

CHAPTER 3. STABILITY ϕ1 (|x|) ≤ V (t, x) ≤ ϕ2 (|x|) V˙ (t, x) ≤ 0

for all |x| ≥ R and t ∈ [0, ∞), then, the solutions of (3.4.1) are u.b. If in addition there exists ϕ3 ∈ K defined on [0, ∞) and (iii) V˙ (t, x) ≤ −ϕ3 (|x|) for all |x| ≥ R and t ∈ [0, ∞) then, the solutions of (3.4.1) are u.u.b. Let us examine statement (ii) of Theorem 3.4.1 where V decrescent and V˙ ≤ 0 imply xe = 0 is u.s. If we remove the restriction of V being decrescent in (ii), we obtain statement (i), i.e., V˙ ≤ 0 implies xe = 0 is stable but not necessarily u.s. Therefore, one might tempted to expect that by removing the condition of V being decrescent in statement (iii), we obtain xe = 0 is a.s., i.e., V˙ < 0 alone implies xe = 0 is a.s. This intuitive conclusion is not true, as demonstrated by a counter example in [206] where a first-order differential equation and a positive definite, nondecrescent function V (t, x) are used to show that V˙ < 0 does not imply a.s. The system (3.4.1) is referred to as nonautonomous. When the function f in (3.4.1) does not depend explicitly on time t, the system is referred to as autonomous. In this case, we write x˙ = f (x)

(3.4.4)

Theorem 3.4.1 to 3.4.3 also hold for (3.4.4) because it is a special case of (3.4.1). In the case of (3.4.4), however, V (t, x) = V (x), i.e., it does not depend explicitly on time t, and all references to the word “decrescent” and “uniform” could be deleted. This is because V (x) is always decrescent and the stability (respectively a.s.) of the equilibrium xe = 0 of (3.4.4) implies u.s. (respectively u.a.s.). For the system (3.4.4), we can obtain a stronger result than Theorem 3.4.2 for a.s. as indicated below. Definition 3.4.24 A set Ω in Rn is invariant with respect to equation (3.4.4) if every solution of (3.4.4) starting in Ω remains in Ω for all t. Theorem 3.4.4 Assume that (3.4.4) possesses unique solutions for all x0 ∈ Rn . Suppose there exists a positive definite and radially unbounded function

3.4. LYAPUNOV STABILITY

113

V (x) : Rn → 7 R+ with continuous first-order derivative with respect to x and V (0) = 0. If (i) V˙ ≤ 0 ∀x ∈ Rn (ii) The origin x = 0 is the only invariant subset of the set n

¯ ¯

Ω = x ∈ Rn ¯V˙ = 0

o

then the equilibrium xe = 0 of (3.4.4) is a.s. in the large. Theorems 3.4.1 to 3.4.4 are referred to as Lyapunov-type theorems. The function V (t, x) or V (x) that satisfies any Lyapunov-type theorem is referred to as Lyapunov function. Lyapunov functions can be also used to predict the instability properties of the equilibrium state xe . Several instability theorems based on the second method of Lyapunov are given in [232]. The following examples demonstrate the use of Lyapunov’s direct method to analyze the stability of nonlinear systems. Example 3.4.2 Consider the system x˙ 1 = x2 + cx1 (x21 + x22 ) x˙ 2 = −x1 + cx2 (x21 + x22 )

(3.4.5)

where c is a constant. Note that xe = 0 is the only equilibrium state. Let us choose V (x) = x21 + x22 as a candidate for a Lyapunov function. V (x) is positive definite, decrescent, and radially unbounded. Its time derivative along the solution of (3.4.5) is V˙ = 2c(x21 + x22 )2

(3.4.6)

If c = 0, then V˙ = 0, and, therefore, xe = 0 is u.s.. If c < 0, then V˙ = −2|c|(x21 +x22 )2 is negative definite, and, therefore, xe = 0 is u.a.s. in the large. If c > 0, xe = 0 is unstable (because in this case V is strictly increasing ∀t ≥ 0), and, therefore, the solutions of (3.4.5) are unbounded [232]. 5

114

CHAPTER 3. STABILITY

Example 3.4.3 Consider the following system describing the motion of a simple pendulum x˙ 1 = x2 (3.4.7) x˙ 2 = −k sin x1 where k > 0 is a constant, x1 is the angle, and x2 the angular velocity. We consider a candidate for a Lyapunov function, the function V (x) representing the total energy of the pendulum given as the sum of the kinetic and potential energy, i.e., Z x1 1 1 V (x) = x22 + k sin η dη = x22 + k(1 − cos x1 ) 2 2 0 V (x) is positive definite and decrescent ∀x ∈ B(π) but not radially unbounded. Along the solution of (3.4.7) we have V˙ = 0 Therefore, the equilibrium state xe = 0 is u.s.

5

Example 3.4.4 Consider the system x˙ 1 = x2 x˙ 2 = −x2 − e−t x1

(3.4.8)

Let us choose the positive definite, decrescent, and radially unbounded function V (x) = x21 + x22 as a Lyapunov candidate. We have V˙ = −2x22 + 2x1 x2 (1 − e−t ) Because for this choice of V function neither of the preceding Lyapunov theorems is applicable, we can reach no conclusion. So let us choose another V function V (t, x) = x21 + et x22 In this case, we obtain

V˙ (t, x) = −et x22

This V function is positive definite, and V˙ is negative semidefinite. Therefore, Theorem 3.4.1 is applicable, and we conclude that the equilibrium state xe = 0 is stable. However, because V is not decrescent, we cannot conclude that the equilibrium state xe = 0 is u.s. 5

3.4. LYAPUNOV STABILITY

115

Example 3.4.5 Let us consider the following differential equations that arise quite often in the analysis of adaptive systems x˙ = −x + φx φ˙ = −x2

(3.4.9)

The equilibrium state is xe = 0, φe = c, where c is any constant, and, therefore, the equilibrium state is not isolated. Let us define φ¯ = φ − c, so that (3.4.9) is transformed into ¯ x˙ = −(1 − c)x + φx ˙φ¯ = −x2

(3.4.10)

We are interested in the stability of the equilibrium point xe = 0, φe = c of (3.4.9), which is equivalent to the stability of xe = 0, φ¯e = 0 of (3.4.10). We choose the positive definite, decrescent, radially unbounded function ¯ = V (x, φ)

x2 φ¯2 + 2 2

(3.4.11)

Then,

¯ = −(1 − c)x2 V˙ (x, φ) If c > 1, then V˙ > 0 for x 6= 0; therefore, xe = 0, φ¯e = 0 is unstable. If, however, c ≤ 1, then xe = 0, φ¯e = 0 is u.s. For c < 1 we have ¯ = −c0 x2 ≤ 0 V˙ (x, φ)

(3.4.12)

where c0 = 1 − c > 0. From Theorem 3.4.3 we can also conclude that the solutions ¯ are u.b. but nothing more. We can exploit the properties of V and V˙ , x(t), φ(t) however, and conclude that x(t) → 0 as t → ∞ as follows. ¯ From (3.4.11) and (3.4.12) we conclude that because V (t) = V (x(t), φ(t)) is bounded from below and is nonincreasing with time, it has a limit, i.e., limt→∞ V (t) = V∞ . Now from (3.4.12) we have Z t Z ∞ V (0) − V∞ 2 lim x dτ = x2 dτ = <∞ t→∞ 0 c0 0 ¯ i.e., x ∈ L2 . Because the solution x(t), φ(t) is u.b., it follows from (3.4.10) that x˙ ∈ L∞ , which together with x ∈ L2 imply (see Lemma 3.2.5) that x(t) → 0 as t → ∞. 5 Example 3.4.6 Consider the differential equation x˙ 1 = −2x1 + x1 x2 + x2 x˙ 2 = −x21 − x1

116

CHAPTER 3. STABILITY 2

2

x x Consider V (x) = 21 + 22 . We have V˙ = −2x21 ≤ 0 and the equilibrium x1e = 0, x2e = 0 is u.s. The set defined in Theorem 3.4.4 is given by

Ω = {x1 , x2 | x1 = 0 } Because x˙ 1 = x2 on Ω, any solution that starts from Ω with x2 6= 0 leaves Ω. Hence, x1 = 0, x2 = 0 is the only invariant subset of Ω. Therefore the equilibrium x1e = 0, x2e = 0 is a.s. in the large. 5

In the proceeding examples, we assume implicitly that the differential equations considered have unique solutions. As indicated in Section 3.4.1, this property is assumed for the general differential equation (3.4.1) on which all definitions and theorems are based. The following example illustrates that if the property of existence of solution is overlooked, an erroneous stability result may be obtained when some of the Lyapunov theorems of this section are used. Example 3.4.7 Consider the second-order differential equation x˙ 1 = −2x1 − x2 sgn(x1 ), x1 (0) = 1 x˙ 2 = |x1 |, x2 (0) = 0 where

½ sgn(x1 ) =

1 if x1 ≥ 0 −1 if x1 < 0

The function V (x1 , x2 ) = x21 + x22 has a time derivative V˙ along the solution of the differential equation that satisfies V˙ (x1 , x2 ) = −4x21 ≤ 0 Hence, according to Theorem 3.4.1, x1 , x2 ∈ L∞ and the equilibrium x1e = 0, x2e = 0 is u.s. Furthermore, we can show that x1 (t) → 0 as t → ∞ as follows: Because V ≥ 0 and V˙ ≤ 0 we have that limt→∞ V (x1 (t), x2 (t)) = V∞ for some V∞ ∈ R+ . Hence, Z t 4x21 (τ )dτ = V (x1 (0), x2 (0)) − V (x1 (t), x2 (t)) 0

which implies that

Z

∞

4 0

x21 (τ )dτ < ∞

3.4. LYAPUNOV STABILITY

117

i.e., x1 ∈ L2 . From x1 , x2 ∈ L∞ we have that x˙ 1 ∈ L∞ , which together with x1 ∈ L2 and Lemma 3.2.5 imply that x1 (t) → 0 as t → ∞. The above conclusions are true provided continuous functions x1 (t), x2 (t) with x1 (0) = 1 and x2 (0) = 0 satisfying the differential equation for all t ∈ [0, ∞) exist. However, the solution of the above differential equation exists only in the time interval t ∈ [0, 1], where it is given by x1 (t) = (1 − t)e−t , x2 (t) = te−t . The difficulty in continuing the solution beyond t = 1 originates from the fact that in a small neighborhood of the point (x1 (1), x2 (1)) = (0, e−1 ), x˙ 1 = −2x1 − x2 sgn(x1 ) < 0 if x1 > 0 and x˙ 1 > 0 if x1 < 0. This causes x˙ 1 to change sign infinitely many times around the point (0, e−1 ), which implies that no continuous functions x1 (t), x2 (t) exist to satisfy the given differential equation past the point t = 1. 5

The main drawback of the Lyapunov’s direct method is that, in general, there is no procedure for finding the appropriate Lyapunov function that satisfies the conditions of Theorems 3.4.1 to 3.4.4 except in the case where (3.4.1) represents a LTI system. If, however, the equilibrium state xe = 0 of (3.4.1) is u.a.s. the existence of a Lyapunov function is assured as shown in [139].

3.4.3

Lyapunov-Like Functions

The choice of an appropriate Lyapunov function to establish stability by using Theorems 3.4.1 to 3.4.4 in the analysis of a large class of adaptive control schemes may not be obvious or possible in many cases. However, a function that resembles a Lyapunov function, but does not possess all the properties that are needed to apply Theorems 3.4.1 to 3.4.4, can be used to establish some properties of adaptive systems that are related to stability and boundedness. We refer to such a function as the Lyapunov-like function. The following example illustrates the use of Lyapunov-like functions. Example 3.4.8 Consider the third-order differential equation x˙ 1 = −x1 − x2 x3 , x1 (0) = x10 x˙ 2 = x1 x3 , x2 (0) = x20 x˙ 3 = x21 , x3 (0) = x30

(3.4.13)

which has the nonisolated equilibrium points in R3 defined by x1 = 0, x2 = constant, x3 = 0 or x1 = 0, x2 = 0, x3 = constant. We would like to analyze the stability

118

CHAPTER 3. STABILITY

properties of the solutions of (3.4.13) by using an appropriate Lyapunov function and applying Theorems 3.4.1 to 3.4.4. If we follow Theorems 3.4.1 to 3.4.4, then we should start with a function V (x1 , x2 , x3 ) that is positive definite in R3 . Instead of doing so let us consider the simple quadratic function V (x1 , x2 ) =

x2 x21 + 2 2 2

which is positive semidefinite in R3 and, therefore, does not satisfy the positive definite condition in R3 of Theorems 3.4.1 to 3.4.4. The time derivative of V along the solution of the differential equation (3.4.13) satisfies V˙ = −x21 ≤ 0

(3.4.14)

which implies that V is a nonincreasing function of time. Therefore, 4

V (x1 (t), x2 (t)) ≤ V (x1 (0), x2 (0)) = V0 and V, x1 , x2 ∈ L∞ . Furthermore, V has a limit as t → ∞, i.e., lim V (x1 (t), x2 (t)) = V∞

t→∞

and (3.4.14) implies that Z

t 0

and

x21 (τ )dτ = V0 − V (t),

Z

∞ 0

∀t ≥ 0

x21 (τ )dτ = V0 − V∞ < ∞

i.e., x1 ∈ L2 . From x1 ∈ L2 we have from (3.3.13) that x3 ∈ L∞ and from x1 , x2 , x3 ∈ L∞ that x˙ 1 ∈ L∞ . Using x˙ 1 ∈ L∞ , x1 ∈ L2 and applying Lemma 3.2.5 we have x1 (t) → 0 as t → ∞. By using the properties of the positive semidefinite function V (x1 , x2 ), we have established that the solution of (3.4.13) is uniformly bounded and x1 (t) → 0 as t → ∞ for any finite initial condition x1 (0), x2 (0), x3 (0). Because the approach we follow resembles the Lyapunov function approach, we are motivated to refer to V (x1 , x2 ) as the Lyapunov-like function. In the above

analysis we also assumed that (3.4.13) has a unique solution. For discussion and analysis on existence and uniqueness of solutions of (3.4.13) the reader is referred to [191]. 5 We use Lyapunov-like functions and similar arguments as in the example above to analyze the stability of a wide class of adaptive schemes considered throughout this book.

3.4. LYAPUNOV STABILITY

3.4.4

119

Lyapunov’s Indirect Method

Under certain conditions, conclusions can be drawn about the stability of the equilibrium of a nonlinear system by studying the behavior of a certain linear system obtained by linearizing (3.4.1) around its equilibrium state. This method is known as the first method of Lyapunov or as Lyapunov’s indirect method and is given as follows [32, 232]: Let xe = 0 be an equilibrium state of (3.4.1) and assume that f (t, x) is continuously differentiable with respect to x for each t ≥ 0. Then in the neighborhood of xe = 0, f has a Taylor series expansion that can be written as x˙ = f (t, x) = A(t)x + f1 (t, x)

(3.4.15)

where A(t) = ∇f |x=0 is referred to as the Jacobian matrix of f evaluated at x = 0 and f1 (t, x) represents the remaining terms in the series expansion. Theorem 3.4.5 Assume that A(t) is uniformly bounded and that lim sup

|x|→0 t≥0

|f1 (t, x)| =0 |x|

Let ze = 0 be the equilibrium of z(t) ˙ = A(t)z(t) The following statements are true for the equilibrium xe = 0 of (3.4.15): (i) If ze = 0 is u.a.s. then xe = 0 is u.a.s. (ii) If ze = 0 is unstable then xe = 0 is unstable (iii) If ze = 0 is u.s. or stable, no conclusions can be drawn about the stability of xe = 0. For a proof of Theorem 3.4.5 see [232]. Example 3.4.9 Consider the second-order differential equation m¨ x = −2µ(x2 − 1)x˙ − kx where m, µ, and k are positive constants, which is known as the Van der Pol oscillator. It describes the motion of a mass-spring-damper with damping coefficient

120

CHAPTER 3. STABILITY

2µ(x2 − 1) and spring constant k, where x is the position of the mass. If we define the states x1 = x, x2 = x, ˙ we obtain the equation x˙ 1

=

x2

x˙ 2

=

−

k 2µ 2 x1 − (x − 1)x2 m m 1

which has an equilibrium at x1e = 0, x2e = 0. The linearization of this system around (0, 0) gives us · ¸ · ¸· ¸ 0 1 z˙1 z1 = 2µ k z˙2 z2 −m m Because m, µ > 0 at least one of the eigenvalues of the matrix A is positive and therefore the equilibrium (0, 0) is unstable. 5

3.4.5

Stability of Linear Systems

Equation (3.4.15) indicates that certain classes of nonlinear systems may be approximated by linear ones in the neighborhood of an equilibrium point or, as often called in practice, operating point. For this reason we are interested in studying the stability of linear systems of the form x(t) ˙ = A(t)x(t)

(3.4.16)

where the elements of A(t) are piecewise continuous for all t ≥ t0 ≥ 0, as a special class of the nonlinear system (3.4.1) or as an approximation of the linearized system (3.4.15). The solution of (3.4.16) is given by [95] x(t; t0 , x0 ) = Φ(t, t0 )x0 for all t ≥ t0 , where Φ(t, t0 ) is the state transition matrix and satisfies the matrix differential equation ∂ Φ(t, t0 ) = A(t)Φ(t, t0 ), ∂t

∀t ≥ t0

Φ(t0 , t0 ) = I Some additional useful properties of Φ(t, t0 ) are (i) Φ(t, t0 ) = Φ(t, τ )Φ(τ, t0 ) ∀t ≥ τ ≥ t0 (semigroup property) (ii) Φ(t, t0 )−1 = Φ(t0 , t) (iii) ∂t∂0 Φ(t, t0 ) = −Φ(t, t0 )A(t0 )

3.4. LYAPUNOV STABILITY

121

Necessary and sufficient conditions for the stability of the equilibrium state xe = 0 of (3.4.16) are given by the following theorems. Theorem 3.4.6 Let kΦ(t, τ )k denote the induced matrix norm of Φ(t, τ ) at each time t ≥ τ . The equilibrium state xe = 0 of (3.4.16) is (i)

stable if and only if the solutions of (3.4.16) are bounded or equivalently 4

c(t0 ) = sup kΦ(t, t0 )k < ∞ t≥t0

(ii)

u.s. if and only if Ã

!

4

c0 = sup c(t0 ) = sup sup kΦ(t, t0 )k t0 ≥0

t0 ≥0

<∞

t≥t0

(iii) a.s. if and only if lim kΦ(t, t0 )k = 0

t→∞

for any t0 ∈ R+ (iv) u.a.s. if and only if there exist positive constants α and β such that kΦ(t, t0 )k ≤ αe−β(t−t0 ) ,

∀t ≥ t0 ≥ 0

(v) e.s. if and only if it is u.a.s. (vi) a.s., u.a.s., e.s. in the large if and only if it is a.s., u.a.s., e.s., respectively. Theorem 3.4.7 [1] Assume that the elements of A(t) are u.b. for all t ∈ R+ . The equilibrium state xe = 0 of the linear system (3.4.16) is u.a.s. if and only if, given any positive definite matrix Q(t), which is continuous in t and satisfies 0 < c1 I ≤ Q(t) ≤ c2 I < ∞ for all t ≥ t0 , the scalar function defined by V (t, x) = x>

Z ∞ t

Φ> (τ, t)Q(τ )Φ(τ, t) dτ x

(3.4.17)

exists (i.e., the integral defined by (3.4.17) is finite for finite values of x and t) and is a Lyapunov function of (3.4.16) with V˙ (t, x) = −x> Q(t)x

122

CHAPTER 3. STABILITY 4R∞ > t Φ (τ, t)Q(τ )Φ(τ, t)dτ

It follows using the properties of Φ(t, t0 ) that P (t) = satisfies the equation

P˙ (t) = −Q(t) − A> (t)P (t) − P (t)A(t)

(3.4.18)

i.e., the Lyapunov function (3.4.17) can be rewritten as V (t, x) = x> P (t)x, where P (t) = P > (t) satisfies (3.4.18). Theorem 3.4.8 A necessary and sufficient condition for the u.a.s of the equilibrium xe = 0 of (3.4.16) is that there exists a symmetric matrix P (t) such that γ1 I ≤ P (t) ≤ γ2 I P˙ (t) + A> (t)P (t) + P (t)A(t) + νC(t)C > (t) ≤ O are satisfied ∀t ≥ 0 and some constant ν > 0, where γ1 > 0, γ2 > 0 are constants and C(t) is such that (C(t), A(t)) is a UCO pair (see Definition 3.3.3). When A(t) = A is a constant matrix, the conditions for stability of the equilibrium xe = 0 of x˙ = Ax (3.4.19) are given by the following theorem. Theorem 3.4.9 The equilibrium state xe = 0 of (3.4.19) is stable if and only if (i) (ii)

All the eigenvalues of A have nonpositive real parts. For each eigenvalue λi with Re{λi } = 0, λi is a simple zero of the minimal polynomial of A (i.e., of the monic polynomial ψ(λ) of least degree such that ψ(A) = O).

Theorem 3.4.10 A necessary and sufficient condition for xe = 0 to be a.s. in the large is that any one of the following conditions is satisfied 1 : (i)

All the eigenvalues of A have negative real parts 1

Note that (iii) includes (ii). Because (ii) is used very often in this book, we list it separately for easy reference.

3.4. LYAPUNOV STABILITY (ii)

123

For every positive definite matrix Q, the following Lyapunov matrix equation A> P + P A = −Q

has a unique solution P that is also positive definite. (iii) For any given matrix C with (C, A) observable, the equation A> P + P A = −C > C has a unique solution P that is positive definite. It is easy to verify that for the LTI system given by (3.4.19), if xe = 0 is stable, it is also u.s. If xe = 0 is a.s., it is also u.a.s. and e.s. in the large. In the rest of the book we will abuse the notation and call the matrix A in (3.4.19) stable when the equilibrium xe = 0 is a.s., i.e., when all the eigenvalues of A have negative real parts and marginally stable when xe = 0 is stable, i.e., A satisfies (i) and (ii) of Theorem 3.4.9. Let us consider again the linear time-varying system (3.4.16) and suppose that for each fixed t all the eigenvalues of the matrix A(t) have negative real parts. In view of Theorem 3.4.10, one may ask whether this condition for A(t) can ensure some form of stability for the equilibrium xe = 0 of (3.4.16). The answer is unfortunately no in general, as demonstrated by the following example given in [232]. Example 3.4.10 Let · A(t) =

−1 + 1.5 cos2 t −1 − 1.5 sin t cos t

1 − 1.5 sin t cos t −1 + 1.5 sin2 t

¸

The eigenvalues of A(t) for each fixed t, √ λ(A(t)) = −.25 ± j.5 1.75 have negative real parts and are also independent of t. Despite this the equilibrium xe = 0 of (3.4.16) is unstable because · .5t ¸ e cos t e−t sin t Φ(t, 0) = −e.5t sin t e−t cos t is unbounded w.r.t. time t.

5

124

CHAPTER 3. STABILITY

Despite Example 3.4.10, Theorem 3.4.10 may be used to obtain some sufficient conditions for a class of A(t), which guarantee that xe = 0 of (3.4.16) is u.a.s. as indicated by the following theorem. Theorem 3.4.11 Let the elements of A(t) in (3.4.16) be differentiable2 and bounded functions of time and assume that (A1) Re{λi (A(t))} ≤ −σs ∀t ≥ 0 and for i = 1, 2, . . . , n where σs > 0 is some constant. ˙ ∈ L2 , then the equilibrium state xe = 0 of (3.4.16) is u.a.s. in (i) If kAk the large. (ii) If any one of the following conditions: (a) (b)

R t+T t R t+T t

˙ )kdτ ≤ µT + α0 , i.e., (kAk) ˙ 12 ∈ S(µ) kA(τ ˙ )k2 dτ ≤ µ2 T + α0 , i.e., kAk ˙ ∈ S(µ2 ) kA(τ

˙ (c) kA(t)k ≤µ is satisfied for some α0 , µ ∈ R+ and ∀t ≥ 0, T ≥ 0, then there exists a µ∗ > 0 such that if µ ∈ [0, µ∗ ), the equilibrium state xe of (3.4.16) is u.a.s. in the large. Proof Using (A1), it follows from Theorem 3.4.10 that the Lyapunov equation A> (t)P (t) + P (t)A(t) = −I

(3.4.20)

has a unique bounded solution P (t) for each fixed t. We consider the following Lyapunov function: V (t, x) = x> P (t)x Then along the solution of (3.4.16) we have V˙ = −|x(t)|2 + x> (t)P˙ (t)x(t)

(3.4.21)

From (3.4.20), P˙ satisfies A> (t)P˙ (t) + P˙ (t)A(t) = −Q(t),

∀t ≥ 0

(3.4.22)

˙ where Q(t) = A˙ > (t)P (t) + P (t)A(t). Because of (A1), it can be verified [95] that Z ∞ > P˙ (t) = eA (t)τ Q(t)eA(t)τ dτ 0 2

The condition of differentiability can be relaxed to Lipschitz continuity.

3.4. LYAPUNOV STABILITY

125

satisfies (3.4.22) for each t ≥ 0, therefore, Z ∞ > kP˙ (t)k ≤ kQ(t)k keA (t)τ kkeA(t)τ kdτ 0

Because (A1) implies that keA(t)τ k ≤ α1 e−α0 τ for some α1 , α0 > 0 it follows that kP˙ (t)k ≤ ckQ(t)k for some c ≥ 0. Then,

˙ kQ(t)k ≤ 2kP (t)kkA(t)k

together with P ∈ L∞ imply that ˙ kP˙ (t)k ≤ βkA(t)k,

∀t ≥ 0

(3.4.23)

for some constant β ≥ 0. Using (3.4.23) in (3.4.21) and noting that P satisfies 0 < β1 ≤ λmin (P ) ≤ λmax (P ) ≤ β2 for some β1 , β2 > 0, we have that 2 ˙ ˙ V˙ (t) ≤ −|x(t)|2 + βkA(t)k|x(t)| ≤ −β2−1 V (t) + ββ1−1 kA(t)kV (t)

therefore, V (t) ≤ e

−

Rt t0

˙ )k)dτ (β2−1 −ββ1−1 kA(τ

V (t0 )

(3.4.24)

Let us prove (ii) first. Using condition (a) in (3.4.24) we have −1

V (t) ≤ e−(β2

−ββ1−1 µ)(t−t0 ) ββ1−1 α0

e

V (t0 )

Therefore, for µ∗ = ββ21β and ∀µ ∈ [0, µ∗ ), V (t) → 0 exponentially fast, which implies that xe = 0 is u.a.s. in the large. To be able to use (b), we rewrite (3.4.24) as Rt −1 ˙ )kdτ kA(τ −β2−1 (t−t0 ) ββ1 t0 V (t) ≤ e e V (t0 ) Using the Schwartz inequality and (b) we have Z

t

t0

µZ ˙ )kdτ kA(τ

t

≤

˙ )k2 dτ kA(τ

¶ 21

t0

√

t − t0

£ 2 ¤1 µ (t − t0 )2 + α0 (t − t0 ) 2 √ √ ≤ µ(t − t0 ) + α0 t − t0

≤

Therefore, V (t) ≤ e−α(t−t0 ) y(t)V (t0 )

126

CHAPTER 3. STABILITY

where α = (1 − γ)β2−1 − ββ1−1 µ, ¤ £ √ √ y(t) = exp −γβ2−1 (t − t0 ) + ββ1−1 α0 t − t0   Ã !2 −1 √ 2 √ ββ α α β β 0 0 2 1 = exp −γβ2−1 + t − t0 − 4γβ12 2γβ2−1 and γ is an arbitrary constant that satisfies 0 < γ < 1. It can be shown that · ¸ β 2 β2 4 y(t) ≤ exp α0 = c ∀t ≥ t0 4γβ12 hence, V (t) ≤ ce−α(t−t0 ) V (t0 ) Choosing µ∗ = β1β(1−γ) , we have that ∀µ ∈ [0, µ∗ ), α > 0 and, therefore, V (t) → 0 2β exponentially fast, which implies that xe = 0 is u.a.s in the large. Since (c) implies (a), the proof of (c) follows directly from that of (a). ˙ ∈ L2 implies (b) with The proof of (i) follows from that of (ii)(b), because kAk µ = 0. 2

In simple words, Theorem 3.4.11 states that if the eigenvalues of A(t) for each fixed time t have negative real parts and if A(t) varies sufficiently slowly most of the time, then the equilibrium state xe = 0 of (3.4.16) is u.a.s.

3.5 3.5.1

Positive Real Functions and Stability Positive Real and Strictly Positive Real Transfer Functions

The concept of PR and SPR transfer functions plays an important role in the stability analysis of a large class of nonlinear systems, which also includes adaptive systems. The definition of PR and SPR transfer functions is derived from network theory. That is, a PR (SPR) rational transfer function can be realized as the driving point impedance of a passive (dissipative) network. Conversely, a passive (dissipative) network has a driving point impedance that is rational and PR (SPR). A passive network is one that does not generate energy, i.e., a network consisting only of resistors, capacitors, and inductors. A dissipative

3.5. POSITIVE REAL FUNCTIONS AND STABILITY

127

network dissipates energy, which implies that it is made up of resistors and capacitors, inductors that are connected in parallel with resistors. In [177, 204], the following equivalent definitions have been given for PR transfer functions by an appeal to network theory. Definition 3.5.1 A rational function G(s) of the complex variable s = σ + jω is called PR if (i) G(s) is real for real s. (ii) Re[G(s)] ≥ 0 for all Re[s] > 0. Lemma 3.5.1 A rational proper transfer function G(s) is PR if and only if (i) G(s) is real for real s. (ii) G(s) is analytic in Re[s] > 0, and the poles on the jω-axis are simple and such that the associated residues are real and positive. (iii) For all real value ω for which s = jω is not a pole of G(s), one has Re[G(jω)] ≥ 0. For SPR transfer functions we have the following Definition. Definition 3.5.2 [177] Assume that G(s) is not identically zero for all s. Then G(s) is SPR if G(s − ²) is PR for some ² > 0. The following theorem gives necessary and sufficient conditions in the frequency domain for a transfer function to be SPR: Theorem 3.5.1 [89] Assume that a rational function G(s) of the complex variable s = σ + jω is real for real s and is not identically zero for all s. Let n∗ be the relative degree of G(s) = Z(s)/R(s) with |n∗ | ≤ 1. Then, G(s) is SPR if and only if (i) G(s) is analytic in Re[s] ≥ 0 (ii) Re[G(jω)] > 0, ∀ω ∈ (−∞, ∞) (iii) (a) When n∗ = 1, lim|ω|→∞ ω 2 Re[G(jω)] > 0. (b) When n∗ = −1, lim|ω|→∞

G(jω) jω

> 0.

128

CHAPTER 3. STABILITY

It should be noted that when n∗ = 0, (i) and (ii) in Theorem 3.5.1 are necessary and sufficient for G(s) to be SPR. This, however, is not true for n∗ = 1 or −1. For example, G(s) = (s + α + β)/[(s + α)(s + β)] α, β > 0 satisfies (i) and (ii) of Theorem 3.5.1, but is not SPR because it does not satisfy (iiia). It is, however, PR. Some useful properties of SPR functions are given by the following corollary. Corollary 3.5.1 (i) G(s) is PR (SPR) if and only if 1/G(s) is PR (SPR) (ii) If G(s) is SPR, then, |n∗ | ≤ 1, and the zeros and poles of G(s) lie in Re[s] < 0 (iii) If |n∗ | > 1, then G(s) is not PR. A necessary condition for G(s) to be PR is that the Nyquist plot of G(jω) lies in the right half complex plane, which implies that the phase shift in the output of a system with transfer function G(s) in response to a sinusoidal input is less than 90◦ . Example 3.5.1 Consider the following transfer functions: s−1 1 , (ii) G2 (s) = (s + 2)2 (s + 2)2 s+3 1 (iii) G3 (s) = , (iv) G4 (s) = (s + 1)(s + 2) s+α (i)

G1 (s) =

Using Corollary 3.5.1 we conclude that G1 (s) is not PR, because 1/G1 (s) is not PR. We also have that G2 (s) is not PR because n∗ > 1. For G3 (s) we have that Re[G3 (jω)] =

6 (s −

ω 2 )2

+ 9ω 2

> 0,

∀ω ∈ (−∞, ∞)

which together with the stability of G3 (s) implies that G3 (s) is PR. Because G3 (s) violates (iii)(a) of Theorem 3.5.1, it is not SPR. The function G4 (s) is SPR for any α > 0 and PR, but not SPR for α = 0. 5

The relationship between PR, SPR transfer functions, and Lyapunov stability of corresponding dynamic systems leads to the development of several

3.5. POSITIVE REAL FUNCTIONS AND STABILITY

129

stability criteria for feedback systems with LTI and nonlinear parts. These criteria include the celebrated Popov’s criterion and its variations [192]. The vital link between PR, SPR transfer functions or matrices and the existence of a Lyapunov function for establishing stability is given by the following lemmas. Lemma 3.5.2 (Kalman-Yakubovich-Popov (KYP) Lemma)[7, 192] Given a square matrix A with all eigenvalues in the closed left half complex plane, a vector B such that (A, B) is controllable, a vector C and a scalar d ≥ 0, the transfer function defined by G(s) = d + C > (sI − A)−1 B is PR if and only if there exist a symmetric positive definite matrix P and a vector q such that A> P + P A = −qq > √ P B − C = ±( 2d)q Lemma 3.5.3 (Lefschetz-Kalman-Yakubovich (LKY) Lemma) [89, 126] Given a stable matrix A, a vector B such that (A, B) is controllable, a vector C and a scalar d ≥ 0, the transfer function defined by G(s) = d + C > (sI − A)−1 B is SPR if and only if for any positive definite matrix L, there exist a symmetric positive definite matrix P , a scalar ν > 0 and a vector q such that A> P + P A = −qq > − νL √ P B − C = ±q 2d The lemmas above are applicable to LTI systems that are controllable. This controllability assumption is relaxed in [142, 172]. Lemma 3.5.4 (Meyer-Kalman-Yakubovich (MKY) Lemma) Given a stable matrix A, vectors B, C and a scalar d ≥ 0, we have the following: If G(s) = d + C > (sI − A)−1 B

130

CHAPTER 3. STABILITY

is SPR, then for any given L = L> > 0, there exists a scalar ν > 0, a vector q and a P = P > > 0 such that A> P + P A = −qq > − νL √ P B − C = ±q 2d In many applications of SPR concepts to adaptive systems, the transfer function G(s) involves stable zero-pole cancellations, which implies that the system associated with the triple (A, B, C) is uncontrollable or unobservable. In these situations the MKY lemma is the appropriate lemma to use. Example 3.5.2 Consider the system y = G(s)u s+3 where G(s) = (s+1)(s+2) . We would like to verify whether G(s) is PR or SPR by using the above lemmas. The system has the state space representation

x˙ = Ax + Bu y = C >x where

· A=

0 −2

1 −3

¸

· , B=

0 1

¸

· , C=

3 1

¸

According to the above lemmas, if G(s) is PR, we have PB = C which implies that

· P =

p1 3

3 1

¸

for some p1 > 0 that should satisfy p1 − 9 > 0 for P to be positive definite. We need to calculate p1 , ν > 0 and q such that A> P + P A = −qq > − νL

(3.5.1)

is satisfied for any L = L> > 0. We have · ¸ 12 11 − p1 − = −qq > − νL 11 − p1 0 · ¸ 12 11 − p1 Now Q = is positive semidefinite for p1 = 11 but is indefinite 11 − p1 0 for p1 6= 11. Because no p1 > 9 exists for which Q is positive definite, no ν > 0 can

3.5. POSITIVE REAL FUNCTIONS AND STABILITY

131

be found to satisfy (3.5.1) for any given L = L> > 0. Therefore, G(s) is not SPR, something we have already verified in Example 3.5.1. For p1 = 11, we can select √ ν = 0 and q = [ 12, 0]> to satisfy (3.5.1). Therefore, G(s) is PR. 5

The KYP and MKY Lemmas are useful in choosing appropriate Lyapunov or Lyapunov-like functions to analyze the stability properties of a wide class of continuous-time adaptive schemes for LTI plants. We illustrate the use of MKY Lemma in adaptive systems by considering the stability properties of the system e˙ = Ac e + Bc θ> ω θ˙ = −Γe1 ω

(3.5.2)

e1 = C > e that often arises in the analysis of a certain class of adaptive schemes. In (3.5.2), Γ = Γ> > 0 is constant, e ∈ Rm , θ ∈ Rn , e1 ∈ R1 and ω = C0> e + C1> em , where em is continuous and em ∈ L∞ . The stability properties of (3.5.2) are given by the following theorem. Theorem 3.5.2 If Ac is a stable matrix and G(s) = C > (sI − Ac )−1 Bc is T ˙ → 0 as t → ∞. SPR, then e, θ, ω ∈ L∞ ; e, θ˙ ∈ L∞ L2 and e(t), e1 (t), θ(t) Proof Because we made no assumptions about the controllability or observability of (Ac , Bc , Cc ), we concentrate on the use of the MKY Lemma. We choose the function V (e, θ) = e> P e + θ> Γ−1 θ where P = P > > 0 satisfies > A> c P + P Ac = −qq − νL P Bc = C

for some vector q, scalar ν > 0 and L = L> > 0 guaranteed by the SPR property of G(s) and the MKY Lemma. Because ω = C0> e + C1> em and em ∈ L∞ can be treated as an arbitrary continuous function of time in L∞ , we can establish that V (e, θ) is positive definite in Rn+m , and V is a Lyapunov function candidate. The time derivative of V along the solution of (3.5.2) is given by V˙ = −e> qq > e − νe> Le ≤ 0

132

CHAPTER 3. STABILITY

which by Theorem 3.4.1 implies that the equilibrium ee , θe = 0 is u.s. and e1 , e, θ ∈ L∞ . Because ω = C0> e + C1> em and em ∈ L∞ , we also have ω ∈ L∞ . By exploiting the properties of V˙ , V further, we can obtain additional properties about the solution of (3.5.2) as follows: From V ≥ 0 and V˙ ≤ 0 we have that lim V (e(t), θ(t)) = V∞

t→∞

and, therefore,

Z ν

∞

e> Ledτ ≤ V0 − V∞

0

where V0 = V (e(0), θ(0)). Because νλmin (L)|e|2 ≤ νe> Le, it follows that e ∈ L2 . T ˙ From (3.5.2) we have |θ(t)| ≤ kΓk|e1 ||ω|. Since ω ∈ L∞ and e1 ∈ L∞ L2 it follows T from Lemma 3.2.3 (ii) that θ˙ ∈ L∞ L2 . Using e, θ, ω ∈ L∞ we obtain e˙ ∈ L∞ , ˙ e → 0 as which together with e ∈ L2 , implies that e(t) → 0 as t → ∞. Hence, e1 , θ, t → ∞, and the proof is complete. 2

The arguments we use to prove Theorem 3.5.2 are standard in adaptive systems and will be repeated in subsequent chapters.

3.5.2

PR and SPR Transfer Function Matrices

The concept of PR transfer functions can be extended to PR transfer function matrices as follows. Definition 3.5.3 [7, 172] An n×n matrix G(s) whose elements are rational functions of the complex variable s is called PR if (i) G(s) has elements that are analytic for Re[s] > 0 (ii) G∗ (s) = G(s∗ ) for Re[s] > 0 (iii) G> (s∗ ) + G(s) is positive semidefinite for Re[s] > 0 where ∗ denotes complex conjugation. Definition 3.5.4 [7] An n × n matrix G(s) is SPR if G(s − ²) is PR for some ² > 0. Necessary and sufficient conditions in the frequency domain for G(s) to be SPR are given by the following theorem.

3.5. POSITIVE REAL FUNCTIONS AND STABILITY

133

Theorem 3.5.3 [215] Consider the n × n rational transfer matrix G(s) = C > (sI − A)−1 B + D

(3.5.3)

where A, B, C, and D are real matrices with appropriate dimensions. Assume that G(s)+G> (−s) has rank n almost everywhere in the complex plane. Then G(s) is SPR if and only if (i)

all elements of G(s) are analytic in Re[s] ≥ 0.

(ii)

G(jω) + G> (−jω) > 0 ∀ω ∈ R.

(iii) (a) lim ω 2 [G(jω) + G> (−jω)] > 0, D + D> ≥ 0 if det[D + D> ] = 0. ω→∞

(b) lim [G(jω) + G> (−jω)] > 0 if det[D + D> ] 6= 0. ω→∞

Necessary and sufficient conditions on the matrices A, B, C, and D in (3.5.3) for G(s) to be PR, SPR are given by the following lemmas which are generalizations of the lemmas in the SISO case to the MIMO case. Theorem 3.5.4 [215] Assume that G(s) given by (3.5.3) is such that G(s)+ G> (−s) has rank n almost everywhere in the complex plane, det(sI − A) has all its zeros in the open left half plane and (A, B) is completely controllable. Then G(s) is SPR if and only if for any real symmetric positive definite matrix L, there exist a real symmetric positive definite matrix P , a scalar ν > 0, real matrices Q and K such that A> P + P A = −QQ> − νL P B = C ± QK K > K = D> + D Lemma 3.5.5 [177] Assume that the transfer matrix G(s) has poles that lie in Re[s] < −γ where γ > 0 and (A, B, C, D) is a minimal realization of G(s). Then, G(s) is SPR if and only if a matrix P = P > > 0, and matrices Q, K exist such that A> P + P A = −QQ> − 2γP P B = C ± QK K > K = D + D>

134

CHAPTER 3. STABILITY du

y∗ + le- C(s) Σ − 6 ye

d + + ? u- ? 0 G (s) y0- l yluΣ Σ 0 + + F (s) ¾

yn

+? d ¾ n Σl +

Figure 3.2 General feedback system. The SPR properties of transfer function matrices are used in the analysis of a class of adaptive schemes designed for multi-input multi-output (MIMO) plants in a manner similar to that of SISO plants.

3.6 3.6.1

Stability of LTI Feedback Systems A General LTI Feedback System

The block diagram of a typical feedback system is shown in Figure 3.2. Here G0 (s) represents the transfer function of the plant model and C(s), F (s) are the cascade and feedback compensators, respectively. The control input u generated from feedback is corrupted by an input disturbance du to form the plant input u0 . Similarly, the output of the plant y0 is corrupted by an output disturbance d to form the actual plant output y. The measured output yn , corrupted by the measurement noise dn , is the input to the compensator F (s) whose output ye is subtracted from the reference (set point) y ∗ to form the error signal e. The transfer functions G0 (s), C(s), F (s) are generally proper and causal, and are allowed to be rational or irrational, which means they may include time delays. The feedback system can be described in a compact I/O matrix form by treating R = [y ∗ , du , d, dn ]> as the input vector, and E = [e, u0 , y, yn ]> and Y = [y0 , ye , u]> as the output vectors, i.e., E = H(s)R,

Y = I1 E + I2 R

(3.6.1)

3.6. STABILITY OF LTI FEEDBACK SYSTEMS

135

where 

H(s) =

1 1 + F CG0

   

1 −F G0 −F −F C 1 −F C −F C CG0 G0 1 −F CG0 CG0 G0 1 1 



0 0 1 0   I1 =  −1 0 0 0  , 0 1 0 0

3.6.2



    



0 0 −1 0   0 0  I2 =  1 0 0 −1 0 0

Internal Stability

Equation (3.6.1) relates all the signals in the system with the external inputs y ∗ , du , d, dn . From a practical viewpoint it is important to guarantee that for any bounded input vector R, all the signals at any point in the feedback system are bounded. This motivates the definition of the so-called internal stability [152, 231]. Definition 3.6.1 The feedback system is internally stable if for any bounded external input R, the signal vectors Y, E are bounded and in addition kY k∞ ≤ c1 kRk∞ ,

kEk∞ ≤ c2 kRk∞

(3.6.2)

for some constants c1 , c2 ≥ 0 that are independent of R. A necessary and sufficient condition for the feedback system (3.6.1) to be internally stable is that each element of H(s) has stable poles, i.e., poles in the open left half s-plane [231]. The concept of internal stability may be confusing to some readers because of the fact that in most undergraduate books the stability of the feedback system is checked by examining the roots of the characteristic equation 1 + F CG0 = 0 The following example is used to illustrate such confusions. Example 3.6.1 Consider G0 (s) =

s−2 1 , C(s) = , F (s) = 1 s−2 s+5

136

CHAPTER 3. STABILITY

for which the characteristic equation 1 + F CG0 = 1 +

(s − 2) 1 =1+ =0 (s + 5)(s − 2) s+5

(3.6.3)

has a single root at s = −6, indicating stability. On the other hand, the transfer matrix H(s) calculated from this example yields     ∗  y e s + 5 −(s + 5)/(s − 2) −(s + 5) −(s + 5)  du   u0  s − 2 1 s + 5 −(s − 2) −(s − 2)   (3.6.4)  =   d   y  s + 6 1 (s + 5)/(s − 2) s+5 1 1 (s + 5)/(s − 2) s+5 s+5 dn yn indicating that a bounded du will produce unbounded e, y, yn , i.e., the feedback system is not internally stable. We should note that in calculating (3.6.3) and (3.6.4), we assume that s+α s+α g(s) and g(s) are the same transfer functions for any constant α. In (3.6.3) the exact cancellation of the pole at s = 2 of G0 (s) by the zero at s = 2 of C(s) led to the wrong stability result, whereas in (3.6.4) such cancellations have no effect on internal stability. This example indicates that internal stability is more complete than the usual stability derived from the roots of 1 + F CG0 = 0. If, however, G0 , C, F are expressed as the ratio of coprime n (s) (s) (s) polynomials, i.e., G0 (s) = np00(s) , C(s) = npcc(s) , F (s) = pff (s) then a necessary and sufficient condition for internal stability [231] is that the roots of the characteristic equation p0 pc pf + n0 nc nf = 0 (3.6.5) are in the open left-half s-plane. For the example under consideration, n0 = 1, p0 = s − 2, nc = s − 2, pc = s + 5, nf = 1, pf = 1 we have (s − 2)(s + 5) + (s − 2) = (s − 2)(s + 6) = 0 which has one unstable root indicating that the feedback is not internally stable.

5

3.6.3

Sensitivity and Complementary Sensitivity Functions

Although internal stability guarantees that all signals in the feedback system are bounded for any bounded external inputs, performance requirements put restrictions on the size of some of the signal bounds. For example, one of the main objectives of a feedback controller is to keep the error between the plant output y and the reference signal y ∗ small in the presence of external inputs, such as reference inputs, bounded disturbances, and noise. Let us consider

3.6. STABILITY OF LTI FEEDBACK SYSTEMS

137

the case where in the feedback system of Figure 3.2, F (s) = 1, du = 0. Using (3.6.1), we can derive the following relationships between the plant output y and external inputs y ∗ , d, dn : y = T0 y ∗ + S0 d − T0 dn where 4

S0 =

1 , 1 + CG0

4

T0 =

(3.6.6)

CG0 1 + CG0

are referred to as the sensitivity function and complementary sensitivity function, respectively. It follows that S0 , T0 satisfy S0 + T0 = 1

(3.6.7)

It is clear from (3.6.6) that for good tracking and output disturbance rejec4

tion, the loop gain L0 = CG0 has to be chosen large so that S0 ≈ 0 and T0 ≈ 1. On the other hand, the suppression of the effect of the measurement noise dn on y requires L0 to be small so that T0 ≈ 0, which from (3.6.7) implies that S0 ≈ 1. This illustrates one of the basic trade-offs in feedback design, which is good reference tracking and disturbance rejection (|L0 | À 1, S0 ≈ 0, T0 ≈ 1) has to be traded off against suppression of measurement noise (|L0 | ¿ 1, S0 ≈ 1, T0 ≈ 0). In a wide class of control problems, y ∗ , d are usually low frequency signals, and dn is dominant only at high frequencies. In this case, C(s) can be designed so that at low frequencies the loop gain L0 is large, i.e., S0 ≈ 0, T0 ≈ 1, and at high frequencies L0 is small, i.e., S0 ≈ 1, T0 ≈ 0. Another reason for requiring the loop gain L0 to be small at high frequencies is the presence of dynamic plant uncertainties whose effect is discussed in later chapters.

3.6.4

Internal Model Principle

In many control problems, the reference input or setpoint y ∗ can be modeled as Qr (s)y ∗ = 0 (3.6.8) 4

d where Qr (s) is a known polynomial, and s = dt is the differential operator. For example, when y ∗ =constant, Qr (s) = s. When y ∗ = t, Qr (s) = s2 and

138

CHAPTER 3. STABILITY

when y ∗ = Asinω0 t for some constants A and ω0 , then Qr (s) = s2 + ω02 , etc. Similarly, a deterministic disturbance d can be modeled as Qd (s)d = 0

(3.6.9)

for some known Qd (s), in cases where sufficient information about d is available. For example, if d is a sinusoidal signal with unknown amplitude and phase but with known frequency ωd then it can be modeled by (3.6.9) with Qd (s) = s2 + ωd2 . The idea behind the internal model principle is that by including the 1 factor Qr (s)Q in the compensator C(s), we can null the effect of y ∗ , d on d (s) the tracking error e = y ∗ − y. To see how this works, consider the feedback system in Figure 3.2 with F (s) = 1, du = dn = 0 and with the reference input y ∗ and disturbance d satisfying (3.6.8) and (3.6.9) respectively for some known polynomials Qr (s), Qd (s). Let us now replace C(s) in Figure 3.2 with C(s) ¯ , C(s) = Q(s)

Q(s) = Qr (s)Qd (s)

(3.6.10)

where C(s) in (3.6.10) is now chosen so that the poles of each element of ¯ H(s) in (3.6.1) with C(s) replaced by C(s) are stable. From (3.6.6) with du = dn = 0 and C replaced by C/Q, we have e = y∗ − y =

1 1+

CG0 Q

y∗ −

1 1+

CG0 Q

d=

1 Q(y ∗ − d) Q + CG0

Because Q = Qr Qd nulls d, y ∗ , i.e., Q(s)d = 0, Q(s)y ∗ = 0, it follows that e=

1 [0] Q + CG0

which together with the stability of 1/(Q + CG0 ) guaranteed by the choice of C(s) imply that e(t) = y ∗ (t) − y(t) tends to zero exponentially fast. The property of exact tracking guaranteed by the internal model principle can also be derived from the values of the sensitivity and complementary sensitivity functions S0 , T0 at the frequencies of y ∗ , d. For example, if y ∗ = sin ω0 t and d = sin ω1 t, i.e., Q(s) = (s2 + ω02 )(s2 + ω12 ) we have S0 =

Q CG0 , T0 = Q + CG0 Q + CG0

3.7. PROBLEMS

139

and S0 (jω0 ) = S0 (jω1 ) = 0, T0 (jω0 ) = T0 (jω1 ) = 1. A special case of the internal model principle is integral control where Q(s) = s which is widely used in industrial control to null the effect of constant set points and disturbances on the tracking error.

3.7 3.1

Problems © ¯ ª (a) Sketch the unit disk defined by the set x ¯x ∈ R2 , |x|p ≤ 1 for (i) p = 1, (ii) p = 2, (iii) p = 3, and (iv) p = ∞. (b) Calculate the L2 norm of the vector function x(t) = [e−2t , e−t ]> by (i) using the | · |2 -norm in R2 and (ii) using the | · |∞ norm in R2 .

3.2 Let y = G(s)u and g(t) be the impulse response of G(s). Consider the induced kyk∞ norm of the operator T defined as kT k∞ = supkuk∞ =1 kuk , where T : u ∈ ∞ L∞ 7→ y ∈ L∞ . Show that kT k∞ = kgk1 . 3.3 Take u(t) = f (t) where f (t), shown in the figure, is a sequence of pulses centered at n with width n13 and amplitude n, where n = 1, 2, . . . , ∞.

f(n) k 1/27 1/8 2 1 1

n 1

2

3

k

(a) Show that u ∈ L1 but u 6∈ L2 and u 6∈ L∞ . (b) If y = G(s)u where G(s) = |y(t)| → 0 as t → ∞.

1 s+1

and u = f , show that y ∈ L1

3.4 Consider the system depicted in the following figure:

T

L∞ and

140

CHAPTER 3. STABILITY u1 + e1 - Σl − 6

-

y2

y1

H1

H2

¾

-

+ e2 ? ¾ u2 Σl +

Let H1 , H2 : L∞e 7→ L∞e satisfy k(H1 e1 )t k ≤ γ1 ke1t k + β1 Z t e−α(t−τ ) γ(τ )ke2τ kdτ + β2 k(H2 e2 )t k ≤ γ2 ke2t k + 0

for all t ≥ 0, where γ1 ≥ 0, γ2 ≥ 0, α > 0, β1 , β2 are constants, γ(t) is a nonnegative continuous function and k(·)t k, k(·)k denote the L∞e , L∞ norm respectively. Let ku1 k ≤ c, ku2 k ≤ c for some constant c ≥ 0, and γ1 γ2 < 1 (small gain). Show that (a) If γ ∈ L2 , then e1 , e2 , y1 , y2 ∈ L∞ . (b) If γ ∈ S(µ), then e1 , e2 , y1 , y2 ∈ L∞ for any µ ∈ [0, µ∗ ), where µ∗ = α2 (1−γ1 γ2 )2 . 2c0 γ 2 1

3.5 Consider the system described by the following equation: x˙ = A(t)x + f (t, x) + u

(3.7.1)

where f (t, x) satisfies |f (t, x)| ≤ γ(t)|x| + γ0 (t) for all x ∈ Rn , t ≥ 0 and f (t, 0) = 0, where γ(t) ≥ 0, γ0 (t) are continuous functions. If the equilibrium ye = 0 of y˙ = A(t)y is u.a.s and γ ∈ S(µ), show that the following statements hold for some µ∗ > 0 and any µ ∈ [0, µ∗ ): (a) u ∈ L∞ and γ0 ∈ S(ν) for any ν ≥ 0 implies x ∈ L∞ (b) u ≡ 0, γ0 ≡ 0 implies that the equilibrium xe = 0 of (3.7.1) is u.a.s in the large (c) u, γ0 ∈ L2 implies that x ∈ L∞ and limt→∞ x(t) = 0 (d) If u ≡ γ0 ≡ 0 then the solution x(t; t0 , x0 ) of (3.7.1) satisfies |x(t; t0 , x0 )| ≤ Ke−β(t−t0 ) |x(t0 )| f or t ≥ t0 ≥ 0 where β = α−c0 µK > 0 for some constant c0 and K, α > 0 are constants in the bound for the state transition matrix Φ(t, τ ) of y˙ = A(t)y, i.e., kΦ(t, τ )k ≤ Ke−α(t−τ ) ,

f or t ≥ τ ≥ t0

3.7. PROBLEMS

141

3.6 Consider the LTI system x˙ = (A + ²B)x where ² > 0 is a scalar. Calculate ²∗ > 0 such that for all ² ∈ [0, ²∗ ), the equilibrium state xe = 0 is e.s. in the large when (a)

· A=

(b)

· A=

−1 0 0 0

10 −2 10 −1

¸

· ,

B=

¸

· ,

5 0

B=

1 0 2 5 −8 2

¸

¸

using (i) the Euclidean norm and (ii) the infinity norm. 3.7 Consider the system given by the following block diagram: r

+ nu G(s)(1 + ∆(s)) Σ − 6 F (s)

y

-

¾

where F (s) is designed such that the closed-loop system is internally stable G 1 when ∆(s) ≡ 0, i.e., 1+F G , 1+F G are stable transfer functions. Derive conditions on G, F using the small gain theorem such that the mapping T : r 7→ y is bounded in L2 for any ∆(s) that satisfies k∆(s)k∞ ≤ δ, where δ > 0 is a given constant. 3.8 Consider the LTI system x˙ = (A + B)x,

x ∈ Rn

where Reλi (A) < 0 and B is an arbitrary constant matrix. Find a bound on kBk for xe = 0 to be e.s. in the large by (a) Using an appropriate Lyapunov function (b) Without the use of a Lyapunov function 3.9 Examine the stability of the equilibrium states of the following differential equations: (a) x˙ = sin t x (b) x˙ = (3t sin t − t)x

142

CHAPTER 3. STABILITY (c) x˙ = a(t)x, where a(t) is a continuous function with a(t) < 0 ∀t ≥ t0 ≥ 0

3.10 Use Lyapunov’s direct method to analyze the stability of the following systems: ½ x˙ 1 = −x1 + x1 x2 (a) x˙ 2 = −γx21 (b) x ¨ + 2x˙ 3 + 2x = 0 · ¸ 1 −3 x (c) x˙ = 2 −5 ½ x˙ 1 = −2x1 + x1 x2 (d) x˙ 2 = −x21 − σx2 3.11 Find the equilibrium state of the scalar differential equation x˙ = −(x − 1)(x − 2)2 and examine their stability properties using (a) Linearization (b) Appropriate Lyapunov functions 3.12 Consider the system described by x˙ 1 = x2 x˙ 2 = −x2 − (k0 + sin t)x1 where k0 > 0 is a constant. Use an appropriate Lyapunov function to investigate stability of the equilibrium states. 3.13 Check the PR and SPR properties of the systems given by the following transfer functions:

(b) G2 (s) =

s+5 (s+1)(s+4) s (s+2)2

(c) G3 (s) =

s−2 (s+3)(s+5)

(d) G4 (s) =

1 s2 +2s+2

(a) G1 (s) =

3.14 Assume that the transfer function G1 (s) and G2 (s) are PR. Check whether the following transfer functions are PR in general: (a) Ga (s) = G1 (s) + G2 (s) (b) Gm (s) = G1 (s)G2 (s) (c) Gf (s) = G1 (s)[1 + G2 (s)]−1

3.7. PROBLEMS

143

Repeat (a), (b), and (c) when G1 (s) and G2 (s) are SPR. 3.15 (a) Let G1 (s) =

1 , s+1

L(s) =

s−1 (s + 1)2

Find a bound3 on ² > 0 for the transfer function G² (s) = G1 (s) + ²L(s) to be PR, SPR. (b) Repeat (a) when G1 (s) =

s+5 , (s + 2)(s + 3)

L(s) = −

1 s+1

Comment on your results. 3.16 Consider the following feedback system: u

+ l Σ + £± 6 BM+ £+ B £ B

-

θ1 s+2

¾ θ2 s+2

θ0

y-

b(s) a(s)

¾

¾

where a(s) = s2 − 3s + 2 and b(s) = s + α. Choose the constants θ0 , θ1 , θ2 such that for α = 5, the transfer function y(s) 1 = Wm (s) = u(s) s+1 What if α = −5? Explain. 3.17 Consider the following system: e˙ = Ae + Bφ sin t, φ˙ = −e1 sin t

e1 = C > e

where φ, e1 ∈ R1 , e ∈ Rn , (A, B) is controllable and Wm (s) = C > (sI −A)−1 B is SPR. Use Lemma 3.5.3 to find an appropriate Lyapunov function to study the stability properties of the system (i.e., of the equilibrium state φe = 0, ee = 0). 3

Find an ²∗ > 0 such that for all ² ∈ [0, ²∗ ), G² (s) is PR, SPR.

Chapter 4

On-Line Parameter Estimation 4.1

Introduction

In Chapter 2, we discussed various types of model representations and parameterizations that describe the behavior of a wide class of dynamic systems. Given the structure of the model, the model response is determined by the values of certain constants referred to as plant or model parameters. In some applications these parameters may be measured or calculated using the laws of physics, properties of materials, etc. In many other applications, this is not possible, and the parameters have to be deduced by observing the system’s response to certain inputs. If the parameters are fixed for all time, their determination is easier, especially when the system is linear and stable. In such a case, simple frequency or time domain techniques may be used to deduce the unknown parameters by processing the measured response data off-line. For this reason, these techniques are often referred to as off-line parameter estimation techniques. In many applications, the structure of the model of the plant may be known, but its parameters may be unknown and changing with time because of changes in operating conditions, aging of equipment, etc., rendering off-line parameter estimation techniques ineffective. The appropriate estimation schemes to use in this case are the ones that provide frequent estimates of the parameters of the plant model by properly processing the plant I/O data on-line. We refer to these scheme 144

4.1. INTRODUCTION

145

as on-line estimation schemes. The purpose of this chapter is to present the design and analysis of a wide class of schemes that can be used for on-line parameter estimation. The essential idea behind on-line estimation is the comparison of the observed system response y(t), with the output of a parameterized model yˆ(θ, t) whose structure is the same as that of the plant model. The parameter vector θ(t) is adjusted continuously so that yˆ(θ, t) approaches y(t) as t increases. Under certain input conditions, yˆ(θ, t) being close to y(t) implies that θ(t) is close to the unknown parameter vector θ∗ of the plant model. The on-line estimation procedure, therefore, involves three steps: In the first step, an appropriate parameterization of the plant model is selected. This is an important step because some plant models are more convenient than others. The second step generating or updating θ(t). The adaptive law is usually a differential equation whose state is θ(t) and is designed using stability considerations or simple optimization techniques to minimize the difference between y(t) and yˆ(θ, t) with respect to θ(t) at each time t. The third and final step is the design of the plant input so that the properties of the adaptive law imply that θ(t) approaches the unknown plant parameter vector θ∗ as t → ∞. This step is more important in problems where the estimation of θ∗ is one of the objectives and is treated separately in Chapter 5. In adaptive control, where the convergence of θ(t) to θ∗ is usually not one of the objectives, the first two steps are the most important ones. In Chapters 6 and 7, a wide class of adaptive controllers are designed by combining the on-line estimation schemes of this chapter with appropriate control laws. Unlike the off-line estimation schemes, the on-line ones are designed to be used with either stable or unstable plants. This is important in adaptive control where stabilization of the unknown plant is one of the immediate objectives. The chapter is organized as follows: We begin with simple examples presented in Section 4.2, which we use to illustrate the basic ideas behind the design and analysis of adaptive laws for on-line parameter estimation. These examples involve plants that are stable and whose states are available for measurement. In Section 4.3, we extend the results of Section 4.2 to plants that may be unstable, and only part of the plant states is available for measurement. We develop a wide class of adaptive laws using a Lyapunov design approach and simple optimization techniques based on the

146

CHAPTER 4. ON-LINE PARAMETER ESTIMATION

gradient method and least squares. The on-line estimation problem where the unknown parameters are constrained to lie inside known convex sets is discussed in Section 4.4. The estimation of parameters that appear in a special bilinear form is a problem that often appears in model reference adaptive control. It is treated in Section 4.5. In the adaptive laws of Sections 4.2 to 4.5, the parameter estimates are generated continuously with time. In Section 4.6, we discuss a class of adaptive laws, referred to as hybrid adaptive laws, where the parameter estimates are generated at finite intervals of time. A summary of the adaptive laws developed is given in Section 4.7 where the main equations of the adaptive laws and their properties are organized in tables. Section 4.8 contains most of the involved proofs dealing with parameter convergence.

4.2

Simple Examples

In this section, we use simple examples to illustrate the derivation and properties of simple on-line parameter estimation schemes. The simplicity of these examples allows the reader to understand the design methodologies and stability issues in parameter estimation without having to deal with the more complex differential equations that arise in the general case.

4.2.1

Scalar Example: One Unknown Parameter

Let us consider the following plant described by the algebraic equation y(t) = θ∗ u(t)

(4.2.1)

where u ∈ L∞ is the scalar input, y(t) is the output, and θ∗ is an unknown scalar. Assuming that u(t), y(t) are measured, it is desired to obtain an estimate of θ∗ at each time t. If the measurements of y, u were noise free, one could simply calculate θ(t), the estimate of θ∗ , as θ(t) =

y(t) u(t)

(4.2.2)

whenever u(t) 6= 0. The division in (4.2.2), however, may not be desirable because u(t) may assume values arbitrarily close to zero. Furthermore, the

4.2. SIMPLE EXAMPLES

147

effect of noise on the measurement of u, y may lead to an erroneous estimate of θ∗ . The noise and computational error effects in (4.2.2) may be reduced by using various other nonrecursive or off-line methods especially when θ∗ is a constant for all t. In our case, we are interested in a recursive or on-line method to generate θ(t). We are looking for a differential equation, which depends on signals that are measured, whose solution is θ(t) and its equilibrium state is θe = θ∗ . The procedure for developing such a differential equation is given below. Using θ(t) as the estimate of θ∗ at time t, we generate the estimated or predicted value yˆ(t) of the output y(t) as yˆ(t) = θ(t)u(t)

(4.2.3)

The prediction or estimation error ²1 , which reflects the parameter uncertainty because θ(t) is different from θ∗ , is formed as the difference between yˆ and y, i.e., ²1 = y − yˆ = y − θu (4.2.4) 4

The dependence of ²1 on the parameter estimation error θ˜ = θ − θ∗ becomes obvious if we use (4.2.1) to substitute for y in (4.2.4), i.e., ˜ ²1 = θ∗ u − θu = −θu

(4.2.5)

The differential equation for generating θ(t) is now developed by minimizing various cost criteria of ²1 with respect to θ using the gradient or Newton’s method. Such criteria are discussed in great detail in Section 4.3. For this example, we concentrate on the simple cost criterion J(θ) =

²21 (y − θu)2 = 2 2

which we minimize with respect to θ. For each time t, the function J(θ) is convex over R1 ; therefore, any local minimum of J is also global and satisfies ∇J(θ) = 0. One can solve ∇J(θ) = −(y − θu)u = 0 for θ and obtain the nonrecursive scheme (4.2.2) or use the gradient method (see Appendix B) to form the recursive scheme θ˙ = −γ∇J(θ) = γ(y − θu)u = γ²1 u,

θ(0) = θ0

(4.2.6)

148

CHAPTER 4. ON-LINE PARAMETER ESTIMATION u ?

y

- θ∗

¥ - × ¦

-

Plant

−? ¾ Σl ²1 ?+ - ×

Z - γ

θ(t)-

6

θ(0)

Figure 4.1 Implementation of the scalar adaptive law (4.2.6). where γ > 0 is a scaling constant, which we refer to as the adaptive gain. In the literature, the differential equation (4.2.6) is referred to as the update law or the adaptive law or the estimator, to name a few. In this book, we refer to (4.2.6) as the adaptive law for updating θ(t) or estimating θ∗ on-line. The implementation of the adaptive law (4.2.6) is shown in Figure 4.1. The stability properties of (4.2.6) are analyzed by rewriting (4.2.6) in terms of the parameter error θ˜ = θ − θ∗ , i.e., ˙ θ˜ = θ˙ − θ˙∗ = γ²1 u − θ˙∗ ˜ and θ∗ is constant, i.e., θ˙∗ = 0, we have Because ²1 = θ∗ u − θu = −θu ˙ ˜ θ˜ = −γu2 θ,

˜ = θ(0) − θ∗ θ(0)

(4.2.7)

We should emphasize that (4.2.7) is used only for analysis. It cannot be used to generate θ(t) because given an initial estimate θ(0) of θ∗ , the initial value ˜ = θ(0) − θ∗ , which is required for implementing (4.2.7) is unknown due θ(0) to the unknown θ∗ . Let us analyze (4.2.7) by choosing the Lyapunov function ˜ = V (θ)

θ˜2 2γ

The time derivative V˙ of V along the solution of (4.2.7) is given by ˙ θ˜> θ˜ V˙ = γ

4.2. SIMPLE EXAMPLES

149

˙ which after substitution of θ˜ from (4.2.7) becomes V˙ = −u2 θ˜2 = −²21 ≤ 0

(4.2.8)

which implies that the equilibrium θ˜e = 0 of (4.2.7) is u.s. Because no further information about u(t) is assumed other than u ∈ L∞ , we cannot guarantee that V˙ < 0 (e.g., take u(t) = 0) and, therefore, cannot establish that θ˜e = 0 is a.s. or e.s. We can, however, use the properties of V, V˙ to establish convergence for the estimation error and other signals in (4.2.6). For example, because V ≥ 0 is a nonincreasing function of time, the ˜ limt→∞ V (θ(t)) = V∞ exists. Therefore, from (4.2.8) we have Z ∞ 0

²21 (τ )dτ

=−

Z ∞ 0

V˙ (τ )dτ = V0 − V∞

˜ where V0 = V (θ(0)), which implies that ²1 ∈ L2 . Now from (4.2.6) and T u ∈ L∞ , we also have that θ˙ ∈ L∞ L2 . Because, as we have shown in Chapter 3, a square integrable function may not have a limit, let alone tend ˙ → 0 as t → ∞ without to zero with time, we cannot establish that ²1 (t), θ(t) additional conditions. If, however, we assume that u˙ ∈ L∞ , then it follows ˜˙ − θ˜u˙ ∈ L∞ ; therefore, from Lemma 3.2.5 we have ²1 (t) → 0 that ²˙1 = −θu as t → ∞, which is implied by ²1 ∈ L2 , ²˙1 ∈ L∞ . This, in turn, leads to ˜˙ ˙ =0 lim θ(t) = lim θ(t)

t→∞

t→∞

(4.2.9)

The conclusion of this analysis is that for any u, u˙ ∈ L∞ , the adaptive law (4.2.6) guarantees that the estimated output yˆ(t) converges to the actual output y(t) and the speed of adaptation (i.e., the rate of change of the ˙ decreases with time and converges to zero asymptotically. parameters θ) One important question to ask at this stage is whether θ(t) converges as t → ∞ and, if it does, is the limt→∞ θ(t) = θ∗ ? ˙ →0 A quick look at (4.2.9) may lead some readers to conclude that θ(t) as t → ∞ implies that θ(t) does converge to a constant. This conclusion is obviously false because the function θ(t) = sin(ln(2 + t)) satisfies (4.2.9) but θ˜2 ˜ has no limit. We have established, however, that V (θ(t)) = 2γ converges to 2 ˜ ˜ V∞ as t → ∞, i.e., limt→∞ θ (t) = 2γV∞ , which implies that θ(t) and, there√ 4 ¯ fore, θ(t) does converge to a constant, i.e., limt→∞ θ(t) = ± 2γV∞ + θ∗ = θ.

150

CHAPTER 4. ON-LINE PARAMETER ESTIMATION

Hence the question which still remains unanswered is whether θ¯ = θ∗ . It is clear from (4.2.1) that for u(t) = 0, a valid member of the class of input signals considered, y(t) = 0 ∀t ≥ 0, which provides absolutely no information about the unknown θ∗ . It is, therefore, obvious that without additional conditions on the input u(t), y(t) may not contain sufficient information about θ∗ for the identification of θ∗ to be possible. For this simple example, we can derive explicit conditions on u(t) that guarantee parameter convergence by considering the closed-form solution of (4.2.7), i.e., Rt 2 −γ u (τ )dτ ˜ ˜ 0 θ(t) = e θ(0) (4.2.10) ˜ does not tend to zero, whereas for For inputs u(t) = 0 or u(t) = e−t , θ(t) 1 ˜ u2 (t) = 1+t , θ(t) tends to zero asymptotically but not exponentially. A ˜ to converge to zero exponentially necessary and sufficient condition for θ(t) fast is that u(t) satisfies Z t+T0 t

u2 (τ )dτ ≥ α0 T0

(4.2.11)

∀t ≥ 0 and for some α0 , T0 > 0 (see Problem 4.1). It is clear that u(t) = 1 1 does not. The property of u satisfies (4.2.11), whereas u(t) = 0 or e−t or 1+t given by (4.2.11) is referred to as persistent excitation (PE) and is crucial in many adaptive schemes where parameter convergence is one of the objectives. The signal u, which satisfies (4.2.11), is referred to be persistently exciting (PE). The PE property of signals is discussed in more detail in Section 4.3. It is clear from (4.2.10), (4.2.11) that the rate of exponential convergence ˜ to zero is proportional to the adaptive gain γ and the constant α0 of θ(t) in (4.2.11), referred to as the level of excitation. Increasing the value of γ will speed up the convergence of θ(t) to θ∗ . A large γ, however, may make the differential equation (4.2.6) “stiff” and, therefore, more difficult to solve numerically. The same methodology as above may be used for the identification of an n-dimensional vector θ∗ that satisfies the algebraic equation y = θ∗> φ

(4.2.12)

where y ∈ R1 , φ ∈ Rn are bounded signals available for measurement. We will deal with this general case in subsequent sections.

4.2. SIMPLE EXAMPLES

4.2.2

151

First-Order Example: Two Unknowns

Consider the following first-order plant x˙ = −ax + bu,

x(0) = x0

(4.2.13)

where the parameters a and b are constant but unknown, and the input u and state x are available for measurement. We assume that a > 0 and u ∈ L∞ so that x ∈ L∞ . The objective is to generate an adaptive law for estimating a and b on-line by using the observed signals u(t) and x(t). As in Section 4.2.1, the adaptive law for generating the estimates a ˆ and ˆb of a and b, respectively, is to be driven by the estimation error ²1 = x − x ˆ

(4.2.14)

where x ˆ is the estimated value of x formed by using the estimates a ˆ and ˆb. The state x ˆ is usually generated by an equation that has the same form as the plant but with a and b replaced by a ˆ and ˆb, respectively. For example, considering the plant equation (4.2.13), we can generate xˆ from x ˆ˙ = −ˆ ax ˆ + ˆbu,

x ˆ(0) = x ˆ0

(P)

Equation (P) is known as the parallel model configuration [123] and the estimation method based on (P) as the output error method [123, 172]. The plant equation, however, may be rewritten in various different forms giving rise to different equations for generating x ˆ. For example, we can add and subtract the term am x, where am > 0 is an arbitrary design constant, in (4.2.13) and rewrite the plant equation as x˙ = −am x + (am − a)x + bu i.e., x=

1 [(am − a)x + bu] s + am

(4.2.15)

Furthermore, we can proceed and rewrite (4.2.15) as x = θ∗> φ h

i>

1 1 where θ∗ = [b, am − a]> , φ = s+a u, s+a x , which is in the form of the m m algebraic equation considered in Section 4.2.1. Therefore, instead of using

152

CHAPTER 4. ON-LINE PARAMETER ESTIMATION

(P), we may also generate x ˆ from x ˆ˙ = −am x ˆ + (am − a ˆ)x + ˆbu that is x ˆ=

1 s+am [(am

−a ˆ)x + ˆbu]

(SP)

by considering the parameterization of the plant given by (4.2.15). Equation (SP) is widely used for parameter estimation and is known as the seriesparallel model [123]. The estimation method based on (SP) is called the equation error method [123, 172]. Various other models that are a combination of (P) and (SP) are generated [123] by considering different parameterizations for the plant (4.2.13). The estimation error ²1 = x − x ˆ satisfies the differential equation ²˙1 = −a²1 + a ˜x ˆ − ˜bu

(P1)

²˙1 = −am ²1 + a ˜x − ˜bu

(SP1)

for model (P) and for model (SP) where 4

a ˜=a ˆ − a,

4ˆ ˜b = b−b

are the parameter errors. Equations (P1) and (SP1) indicate how the parameter error affects the estimation error ²1 . Because a, am > 0, zero parameter error, i.e., a ˜ = ˜b = 0, implies that ²1 converges to zero exponentially. Be˜ cause a ˜, b are unknown, ²1 is the only measured signal that we can monitor in practice to check the success of estimation. We should emphasize, however, that ²1 → 0 does not imply that a ˜, ˜b → 0 unless some PE properties are satisfied by x ˆ, x, u as we will demonstrate later on in this section. We should also note that ²1 cannot be generated from (P1) and (SP1) because a ˜ and ˜b are unknown. Equations (P1) and (SP1) are, therefore, only used for the purpose of analysis. Let us now use the error equation (SP1) to derive the adaptive laws for estimating a and b. We assume that the adaptive laws are of the form a ˆ˙ = f1 (²1 , x ˆ, x, u),

ˆb˙ = f2 (²1 , x ˆ, x, u)

(4.2.16)

where f1 and f2 are functions of measured signals, and are to be chosen so that the equilibrium state a ˆe = a,

ˆbe = b,

²1e = 0

(4.2.17)

4.2. SIMPLE EXAMPLES

153

of the third-order differential equation described by (SP1) (where x ∈ L∞ is treated as an independent function of time) and (4.2.16) is u.s., or, if possible, u.a.s., or, even better, e.s. We choose f1 , f2 so that a certain function V (²1 , a ˜, ˜b) and its time derivative V˙ along the solution of (SP1), (4.2.16) are such that V qualifies as a Lyapunov function that satisfies some of the conditions given by Theorems 3.4.1 to 3.4.4 in Chapter 3. We start by considering the quadratic function 1 V (²1 , a ˜, ˜b) = (²21 + a ˜2 + ˜b2 ) 2

(4.2.18)

which is positive definite, decrescent, and radially unbounded in R3 . The time derivative of V along any trajectory of (SP1), (4.2.16) is given by V˙ = −am ²21 + a ˜x²1 − ˜bu²1 + a ˜f1 + ˜bf2

(4.2.19)

˙ ˙ and is evaluated by using the identities a ˆ˙ = a ˜˙ , ˆb = ˜b, which hold because a and b are assumed to be constant. If we choose f1 = −²1 x, f2 = ²1 u, we have

and (4.2.16) becomes

V˙ = −am ²21 ≤ 0

(4.2.20)

˙ a ˆ˙ = −²1 x, ˆb = ²1 u

(4.2.21)

where ²1 = x − x ˆ and x ˆ is generated by (SP). Applying Theorem 3.4.1 to (4.2.18) and (4.2.20), we conclude that V is a Lyapunov function for the system (SP1), (4.2.16) where x and u are treated as independent bounded functions of time and the equilibrium given by (4.2.17) is u.s. Furthermore, the trajectory ²1 (t), a ˆ(t), ˆb(t) is bounded for all t ≥ 0. Because ²1 = x − x ˆ and x ∈ L∞ we also have that x ˆ ∈ L∞ ; therefore, all signals in (SP1) and (4.2.21) are uniformly bounded. As in the example given in Section 4.2.1, (4.2.18) and (4.2.20) imply that lim V (²1 (t), a ˜(t), ˜b(t)) = V∞ < ∞

t→∞

and, therefore, Z ∞ 0

²21 (τ )dτ

1 =− am

Z ∞ 0

1 V˙ dτ = (V0 − V∞ ) am

154

CHAPTER 4. ON-LINE PARAMETER ESTIMATION

where V0 = V (²1 (0), a ˜(0), ˜b(0)), i.e., ²1 ∈ L2 . Because u, a ˜, ˜b, x, ²1 ∈ L∞ , it follows from (SP1) that ²˙1 ∈ L∞ , which, together with ²1 ∈ L2 , implies that ˙ ²1 (t) → 0 as t → ∞, which, in turn, implies that a ˆ˙ (t), ˆb(t) → 0 as t → ∞. ˙ It is worth noting that ²1 (t), a ˆ˙ (t), ˆb(t) → 0 as t → ∞ do not imply that a ˜ and ˜b converge to any constant let alone to zero. As in the example of Section 4.2.1, we can use (4.2.18) and (4.2.20) and establish that lim (˜ a2 (t) + ˜b2 (t)) = 2V∞

t→∞

which again does not imply that a ˜ and ˜b have a limit, e.g., take a ˜(t) =

p

p √ √ 2V∞ sin 1 + t, ˜b(t) = 2V∞ cos 1 + t

The failure to establish parameter convergence may motivate the reader to question the choice of the Lyapunov function given by (4.2.18) and of the functions f1 , f2 in (4.2.19). The reader may argue that perhaps for some other choices of V and f1 , f2 , u.a.s could be established for the equilibrium (4.2.17) that will automatically imply that a ˜, ˜b → 0 as t → ∞. Since given a differential equation, there is no procedure for finding the appropriate Lyapunov function to establish stability in general, this argument appears to be quite valid. We can counteract this argument, however, by applying simple intuition to the plant equation (4.2.13). In our analysis, we put no restriction on the input signal u, apart from u ∈ L∞ , and no assumption is made about the initial state x0 . For u = 0, an allowable input in our analysis, and x0 = 0, no information can be extracted about the unknown parameters a, b from the measurements of x(t) = 0, u(t) = 0, ∀t ≥ 0. Therefore, no matter how intelligent an adaptive law is, parameter error convergence to zero cannot be achieved when u = 0 ∀t ≥ 0. This simplistic explanation demonstrates that additional conditions have to be imposed on the input signal u to establish parameter error convergence to zero. Therefore, no matter what V and f1 , f2 we choose, we can not establish u.a.s. without imposing conditions on the input u. These conditions are similar to those imposed on the input u in Section 4.2.1, and will be discussed and analyzed in Chapter 5. In the adaptive law (4.2.21), the adaptive gains are set equal to 1. A similar adaptive law with arbitrary adaptive gains γ1 , γ2 > 0 is derived by

4.2. SIMPLE EXAMPLES considering

155 Ã

1 2 a ˜2 ˜b2 V (²1 , a ˜, ˜b) = ²1 + + 2 γ1 γ2

!

instead of (4.2.18). Following the same procedure as before we obtain ˙ a ˆ˙ = −γ1 ²1 x, ˆb = γ2 ²1 u where γ1 , γ2 > 0 are chosen appropriately to slow down or speed up adaptation. Using (4.2.18) with model (P1) and following the same analysis as with model (SP1), we obtain ˙ a ˆ˙ = −²1 x ˆ, ˆb = ²1 u

(4.2.22)

and V˙ = −a²21 ≤ 0 Hence, the same conclusions as with (4.2.21) are drawn for (4.2.22). We should note that V˙ for (P1) depends on the unknown a, whereas for (SP1) it depends on the known design scalar am . Another crucial difference between model (P) and (SP) is their performance in the presence of noise, which becomes clear after rewriting the adaptive law for a ˆ in (4.2.21), (4.2.22) as a ˆ˙ = −(x − x ˆ)ˆ x=x ˆ2 − xˆ x

(P)

a ˆ˙ = −(x − x ˆ)x = −x2 + xˆ x

(SP)

If the measured plant state x is corrupted by some noise signal v, i.e., x is replaced by x + v in the adaptive law, it is clear that for the model (SP), a ˆ˙ will depend on v 2 and v, whereas for model (P) only on v. The effect of noise (v 2 ) may result in biased estimates in the case of model (SP), whereas the quality of estimation will be less affected in the case of model (P). The difference between the two models led some researchers to the development of more complicated models that combine the good noise properties of the parallel model (P) with the design flexibility of the series-parallel model (SP) [47, 123].

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

Simulations We simulate the parallel and series-parallel estimators and examine the effects of the input signal u, the adaptive gain and noise disturbance on their b performance. For simplicity, we consider a first-order example y = s+a u with two unknown parameters a and b. Two adaptive estimators ˙ a ˆ˙ = −²1 x ˆ, ˆb = ²1 u x ˆ˙ = −ˆ ax ˆ + ˆbu, ²1 = x − x ˆ and ˙ a ˆ˙ = −²1 x, ˆb = ²1 u x ˆ˙ = −am x ˆ + (am − a ˆ)x + ˆbu, ²1 = x − x ˆ based on the parallel and series-parallel model, respectively, are simulated with a = 2 and b = 1. The results are given in Figures 4.2 and Figure 4.3, respectively. Plots (a) and (b) in Figure 4.2 and 4.3 give the time response of the estimated parameters when the input u = sin 5t, and the adaptive gain γ = 1 for (a) and γ = 5 for (b). Plots (c) in both figures give the results of estimation for a step input, where persistent excitation and, therefore, parameter convergence are not guaranteed. Plots (d) show the performance of the estimator when the measurement x(t) is corrupted by d(t) = 0.1n(t), where n(t) is a normally distributed white noise. It is clear from Figures 4.2 (a,b) and Figure 4.3 (a,b) that the use of a larger value of the adaptive gain γ led to a faster convergence of a ˆ and ˆb to their true values. The lack of parameter convergence to the true values in Figure 4.2 (c), 4.3 (c) is due to the use of a non-PE input signal. As expected, the parameter estimates are more biased in the case of the seriesparallel estimator shown in Figure 4.3 (d) than those of the parallel one shown in Figure 4.2 (d).

4.2.3

Vector Case

Let us extend the example of Section 4.2.2 to the higher-order case where the plant is described by the vector differential equation x˙ = Ap x + Bp u

(4.2.23)

4.2. SIMPLE EXAMPLES

157

3

3

aˆ

aˆ

2

2

bˆ

bˆ

1

0

1

0

50

100 (a)

150

0

200

0

50

sec

3

100 (b)

150

200 sec

3

aˆ aˆ

2

bˆ

bˆ

1

0

2

0

10

1

20 (c)

0

30

0

50

sec

100 (d)

150

200 sec

Figure 4.2 Simulation results of the parallel estimator. (a) u = sin 5t, γ = 1, no measurement noise; (b) u = sin 5t, γ = 5, no measurement noise; (c) u =unit step function, γ = 1, no measurement noise; (d) u = sin 5t, γ = 1, output x is corrupted by d(t) = 0.1n(t), where n(t) is a normally distributed white noise.

where the state x ∈ Rn and input u ∈ Rr are available for measurement, Ap ∈ Rn×n , Bp ∈ Rn×r are unknown, Ap is stable, and u ∈ L∞ . As in the scalar case, we form the parallel model ˆp u, x x ˆ˙ = Aˆp x ˆ+B ˆ ∈ Rn

(P)

ˆp (t) are the estimates of Ap , Bp at time t to be generated by where Aˆp (t), B

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

3

3

aˆ

aˆ

2

2

bˆ 1

0

bˆ

1

0

100

200

0

300

0

100

sec

(a) 3

200

300 sec

(b) 3

aˆ 2

2

bˆ

aˆ 1

0

1

bˆ 0

10

20 (c)

0

30

0

100

sec

200 (d)

300 sec

Figure 4.3 Simulation results of the series-parallel estimator. (a) u = sin 5t, γ = 1, no measurement noise; (b) u = sin 5t, γ = 5, no measurement noise; (c) u =unit step function, γ = 1, no measurement noise; (d) u = sin 5t, γ = 1, output x is corrupted by d(t) = 0.1n(t), where n(t) is the normally distributed white noise.

an adaptive law, and x ˆ(t) is the estimate of the vector x(t). Similarly, by considering the plant parameterization x˙ = Am x + (Ap − Am )x + Bp u where Am is an arbitrary stable matrix, we define the series-parallel model as ˆp u x ˆ˙ = Am x ˆ + (Aˆp − Am )x + B

(SP)

4.2. SIMPLE EXAMPLES

159

The estimation error vector ²1 defined as 4

²1 = x − x ˆ satisfies ˜p u ²˙1 = Ap ²1 − A˜p x ˆ−B

(P1)

˜p u ²˙1 = Am ²1 − A˜p x − B

(SP1)

for model (P) and 4 ˆ 4 ˜p = Bp − Bp . for model (SP), where A˜p = Aˆp − Ap , B Let us consider the parallel model design and use (P1) to derive the adaptive law for estimating the elements of Ap , Bp . We assume that the adaptive law has the general structure

˙ ˆ˙ p = F2 (²1 , x, x Aˆp = F1 (²1 , x, x ˆ, u), B ˆ, u)

(4.2.24)

where F1 and F2 are functions of known signals that are to be chosen so that the equilibrium ˆpe = Bp , ²1e = 0 Aˆpe = Ap , B of (P1), (4.2.24) has some desired stability properties. We start by considering the function ˜p ) = V (²1 , A˜p , B

²> 1 P ²1

+ tr

Ã > ! A˜p P A˜p

γ1

Ã

˜p> P B ˜p B + tr γ2

!

(4.2.25)

where tr(A) denotes the trace of a matrix A, γ1 , γ2 > 0 are constant scalars, and P = P > > 0 is chosen as the solution of the Lyapunov equation P Ap + A> p P = −I

(4.2.26)

whose existence is guaranteed by the stability of Ap (see Theorem 3.4.10). The time derivative V˙ of V along the trajectory of (P1), (4.2.24) is given by V˙

> = ²˙> 1 P ²1 + ²1 P ²˙1









> ˙˜ ˙˜> ˜ ˜ ˜˙ p  ˜˙ ˜p> P B A˜> B  Ap P Ap  B p P Bp p P Ap  +tr  + +  + tr   γ1 γ1 γ2 γ2

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

˙˜ , B ˜˙ p becomes which after substituting for ²˙1 , A p !

Ã

˜ > P F2 B A˜> p p P F1 V˙ = 2 +2 γ1 γ2 (4.2.27) We use the following properties of trace to manipulate (4.2.27): > > ˜ ˜ ²> ˆ − 2²> 1 (P Ap + Ap P )²1 − 2²1 P Ap x 1 P Bp u + tr

(i) tr(AB) = tr(BA) (ii) tr(A + B) = tr(A) + tr(B) for any A, B ∈ Rn×n (iii) tr(yx> ) = x> y for any x, y ∈ Rn×1 We have ˜ ˆ=x ˜> ²> ˆ> A˜> ˆ> ) 1 P Ap x p P ²1 = tr(Ap P ²1 x > ˜ ˜> ²> 1 P Bp u = tr(Bp P ²1 u )

and, therefore, V˙ = −²> 1 ²1 + 2tr

Ã > A˜p P F1

γ1

˜ p P F2 B ˜p> P ²1 u> − A˜> ˆ> + −B p P ²1 x γ2

!

(4.2.28)

The obvious choice for F1 , F2 to make V˙ negative is ˙ Aˆp = F1 = γ1 ²1 x ˆ> ,

ˆ˙ p = F2 = γ2 ²1 u> B

(4.2.29)

In the case of the series-parallel model, we choose ˜ p ) = ²> V (²1 , A˜p , B 1 P ²1 + tr

! Ã > A˜p P A˜p

γ1

Ã

˜ >P B ˜p B p + tr γ2

!

(4.2.30)

where P = P > > 0 is the solution of the Lyapunov equation A> m P + P Am = −I

(4.2.31)

By following the same procedure as in the case of the parallel model, we obtain ˙ ˆ˙ p = γ2 ²1 u> Aˆp = γ1 ²1 x> , B (4.2.32)

4.2. SIMPLE EXAMPLES

161

Remark 4.2.1 If instead of (4.2.30), we choose ³

´

³

˜p ) = ²> ˜> ˜ ˜> ˜ V (²1 , A˜p , B 1 P ²1 + tr Ap Ap + tr Bp Bp

´

(4.2.33)

where P = P > > 0 satisfies (4.2.31), we obtain ˙ Aˆp = P ²1 x> ,

ˆ˙ p = P ²1 u> B

(4.2.34)

In this case P is the adaptive gain matrix and is calculated using (4.2.31). Because Am is a known stable matrix, the calculation of P is possible. It should be noted that if we use the same procedure for the parallel model, we will end up with an adaptive law that depends on P that satisfies the Lyapunov equation (4.2.26) for Ap . Because Ap is unknown, P cannot be calculated; therefore, the adaptive laws corresponding to (4.2.34) for the parallel model are not implementable. The time derivative V˙ of V in both the parallel and series-parallel estimators satisfies V˙ = −²> 1 ²1 ≤ 0 ˆpe = Bp , ²1e = 0 of the respecwhich implies that the equilibrium Aˆpe = Ap , B tive equations is u.s. Using arguments similar to those used in Section 4.2.2 we establish that ²1 ∈ L2 , ²˙1 ∈ L∞ and that ˙ ˆ˙ p (t)k → 0 as t → ∞ |²1 (t)| → 0, kAˆp (t)k → 0, kB ˆp to their true values Ap , Bp , respectively, The convergence properties of Aˆp , B depend on the properties of the input u. As we will discuss in Chapter 5, if u belongs to the class of sufficiently rich inputs, i.e., u has enough frequencies to excite all the modes of the plant, then the vector [x> , u> ]> is PE and ˆp converge to Ap , Bp , respectively, exponentially fast. guarantees that Aˆp , B

4.2.4

Remarks

(i) In this section we consider the design of on-line parameter estimators for simple plants that are stable, whose states are accessible for measurement and whose input u is bounded. Because no feedback is used and the plant is not disturbed by any signal other than u, the stability of the plant is not an issue. The main concern, therefore, is the stability properties of the estimator or adaptive law that generates the on-line estimates for the unknown plant parameters.

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

(ii) For the examples considered, we are able to design on-line parameter estimation schemes that guarantee that the estimation error ²1 converges to zero as t → ∞, i.e., the predicted state x ˆ approaches that of the plant as t → ∞ and the estimated parameters change more and more slowly as time increases. This result, however, is not sufficient to establish parameter convergence to the true parameter values unless the input signal u is sufficiently rich. To be sufficiently rich, u has to have enough frequencies to excite all the modes of the plant. (iii) The properties of the adaptive schemes developed in this section rely on the stability of the plant and the boundedness of the plant input u. Consequently, they may not be appropriate for use in connection with control problems where u is the result of feedback and is, therefore, no longer guaranteed to be bounded a priori . In the following sections, we develop on-line parameter estimation schemes that do not rely on the stability of the plant and the boundedness of the plant input.

4.3

Adaptive Laws with Normalization

In Section 4.2, we used several simple examples to illustrate the design of various adaptive laws under the assumption that the full state of the plant is available for measurement, the plant is stable, and the plant input is bounded. In this section, we develop adaptive laws that do not require the plant to be stable or the plant input to be bounded a priori. Such adaptive laws are essential in adaptive control, to be considered in later chapters, where the stability of the plant and the boundedness of the plant input are properties to be proven and, therefore, cannot be assumed to hold a priori. We begin with simple examples of plants whose inputs and states are not restricted to be bounded and develop adaptive laws using various approaches. These results are then generalized to a wider class of higher-order plants whose output rather than the full state vector is available for measurement.

4.3.1

Scalar Example

Let us consider the simple plant given by the algebraic equation y(t) = θ∗ u(t)

(4.3.1)

4.3. ADAPTIVE LAWS WITH NORMALIZATION

163

where u and, therefore, y are piecewise continuous signals but not necessarily bounded, and θ∗ is to be estimated by using the measurements of y and u. As in Section 4.2.1, we can generate the estimate yˆ of y and the estimation error ²1 as yˆ = θu ²1 = y − yˆ = y − θu where θ(t) is the estimate of θ∗ at time t. Because u and y are not guaranteed to be bounded, the minimization problem (y − θu)2 min J(θ) = min θ θ 2 is ill posed and, therefore, the procedure of Section 4.2.1 where u, y ∈ L∞ does not extend to the case where u, y 6∈ L∞ . This obstacle is avoided by dividing each side of (4.3.1) with some function referred to as the normalizing signal m > 0 to obtain y¯ = θ∗ u ¯ (4.3.2) where

y u , u ¯= m m are the normalized values of y and u, respectively, and m2 = 1 + n2s . The u signal ns is chosen so that m ∈ L∞ . A straightforward choice for ns is 2 2 ns = u, i.e., m = 1 + u . Because u ¯, y¯ ∈ L∞ , we can follow the procedure of Section 4.2.1 and develop an adaptive law based on (4.3.2) rather than on (4.3.1) as follows: The estimated value yˆ¯ of y¯ is generated as y¯ =

yˆ¯ = θ¯ u and the estimation error as ²¯1 = y¯ − yˆ¯ = y¯ − θ¯ u ˆ¯ = yˆ/m. The adaptive law for θ is now It is clear that ²¯1 = ²m1 = y−θu m and y developed by solving the well-posed minimization problem min J(θ) = min θ

θ

(¯ y − θ¯ u)2 (y − θu)2 = min θ 2 2m2

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

Using the gradient method we obtain θ˙ = γ¯ ²1 u ¯,

γ>0

(4.3.3)

or in terms of the unnormalized signals ²1 u θ˙ = γ 2 m

(4.3.4)

where m may be chosen as m2 = 1 + u2 . For clarity of presentation, we rewrite (4.3.3) and (4.3.4) as θ˙ = γ²u

(4.3.5)

where

²1 m2 We refer to ² as the normalized estimation error. This notation enables us to unify results based on different approaches and is adopted throughout the book. Let us now analyze (4.3.5) by rewriting it in terms of the parameter error ˜ ˜ ˙ θ˜ = θ − θ∗ . Using θ˜ = θ˙ and ² = ²1 /m2 = − θu2 = − θu¯ , we have ²=

m

m

˙ θ˜ = −γ u ¯2 θ˜

(4.3.6)

We propose the Lyapunov function ˜2 ˜ = θ V (θ) 2γ whose time derivative along the solution of (4.3.6) is given by V˙ = −θ˜2 u ¯2 = −²2 m2 ≤ 0 u ¯ ˜ θ ∈ L∞ and ²m ∈ L2 . Because θ, ˜u Hence, θ, ¯ ∈ L∞ , it follows from ² = −θ˜m that ², ²m ∈ L∞ . If we now rewrite (4.3.5) as θ˙ = γ²m¯ u, it follows from T T d ˜˙ u − θ˜u ˙ ¯˙ ²m ∈ L∞ L2 and u ¯ ∈ L∞ that θ ∈ L∞ L2 . Since dt ²m = −θ¯ ˙˜ ˜ d and θ, θ, u ¯ ∈ L∞ , it follows that for u ¯˙ ∈ L∞ we have dt (²m) ∈ L∞ , which, together with ²m ∈ L2 , implies that ²m → 0 as t → ∞. This, in turn, ˙ implies that θ˜ → 0 as t → ∞.

4.3. ADAPTIVE LAWS WITH NORMALIZATION

165

The significance of this example is that even in the case of unbounded y, u we are able to develop an adaptive law that guarantees bounded parameter estimates and a speed of adaptation that is bounded in an L2 and L∞ sense T (i.e., θ˙ ∈ L∞ L2 ). When ns = 0, i.e., m = 1 the adaptive law (4.3.5) becomes the unnormalized one considered in Section 4.2. It is obvious that for m = 1, in (4.3.5), i.e., ² = ²1 , we can still establish parameter boundedness, but we can not guarantee boundedness for θ˙ in an Lp sense unless u ∈ L∞ . As we will demonstrate in Chapter 6, the property that θ˙ ∈ L2 is crucial for stability when adaptive laws of the form (4.3.5) are used with control laws based on the certainty equivalence principle to stabilize unknown plants.

4.3.2

First-Order Example

Let us now consider the same plant (4.2.15) given in Section 4.2.2, i.e., x=

1 [(am − a)x + u] s + am

(4.3.7)

where for simplicity we assume that b = 1 is known, u is piecewise continuous but not necessarily bounded, and a may be positive or negative. Unlike the example in Section 4.2.2, we make no assumptions about the boundedness of x and u. Our objective is to develop an adaptive law for estimating a on-line using the measurements of x, u. If we adopt the approach of the example in Section 4.2.2, we will have x ˆ=

1 [(am − a ˆ)x + u], ²1 = x − x ˆ s + am

(4.3.8)

and a ˆ˙ = −²1 x.

(4.3.9)

Let us now analyze (4.3.8) and (4.3.9) without using any assumption about the boundedness of x, u. We consider the estimation error equation ²˙1 = −am ²1 + a ˜x

(4.3.10)

4

where a ˜=a ˆ − a and propose the same function V =

²21 a ˜2 + 2 2

(4.3.11)

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

as in Section 4.2.2. The time derivative of V along (4.3.9) and (4.3.10) is given by V˙ = −am ²21 ≤ 0 (4.3.12) Because x is not necessarily bounded, it cannot be treated as an independent bounded function of time in (4.3.10) and, therefore, (4.3.10) cannot be decoupled from (4.3.8). Consequently, (4.3.8) to (4.3.10) have to be considered and analyzed together in R3 , the space of ²1 , a ˜, x ˆ. The chosen function V 3 in (4.3.11) is only positive semidefinite in R , which implies that V is not a Lyapunov function; therefore, Theorems 3.4.1 to 3.4.4 cannot be applied. V is, therefore, a Lyapunov-like function, and the properties of V , V˙ allow us to draw some conclusions about the behavior of the solution ²1 (t), a ˜(t) without having to apply the Lyapunov Theorems 3.4.1 to 3.4.4. From V ≥ 0 and ˜ ∈ L∞ , V˙ = −am ²21 ≤ 0 we conclude that V ∈ L∞ , which implies that ²1 , a and ²1 ∈ L2 . Without assuming x ∈ L∞ , however, we cannot establish any bound for a ˜˙ in an Lp sense. As in Section 4.3.1, let us attempt to use normalization and modify (4.3.9) to achieve bounded speed of adaptation in some sense. The use of normalization is not straightforward in this case because of the dynamics 1 introduced by the transfer function s+a , i.e., dividing each side of (4.3.7) m by m may not help because ·

x 1 x u 6= (am − a) + m s + am m m

¸

For this case, we propose the error signal ²=x−x ˆ−

1 1 ²n2s = (˜ ax − ²n2s ) s + am s + am

(4.3.13)

i.e., ²˙ = −am ² + a ˜x − ²n2s where ns is a normalizing signal to be designed. Let us now use the error equation (4.3.13) to develop an adaptive law for a ˆ. We consider the Lyapunov-like function V =

²2 a ˜2 + 2 2

(4.3.14)

4.3. ADAPTIVE LAWS WITH NORMALIZATION

167

whose time derivative along the solution of (4.3.13) is given by V˙ = −am ²2 − ²2 n2s + a ˜²x + a ˜a ˜˙ Choosing a ˜˙ = a ˆ˙ = −²x

(4.3.15)

we have V˙ = −am ²2 − ²2 n2s ≤ 0 which together with (4.3.14) imply V, ², a ˜ ∈ L∞ and ², ²ns ∈ L2 . If we now write (4.3.15) as x a ˜˙ = −²m m x where m2 = 1 + n2s and choose ns so that m ∈ L∞ , then ²m ∈ L2 (because ˙ ², ²ns ∈ L2 ) implies that a ˜ ∈ L2 . A straightforward choice for ns is ns = x, 2 2 i.e., m = 1 + x . The effect of ns can be roughly seen by rewriting (4.3.13) as

²˙ = −am ² − ²n2s + a ˜x

(4.3.16)

and solving for the “quasi” steady-state response ²s =

a ˜x am + n2s

(4.3.17)

obtained by setting ²˙ ≈ 0 in (4.3.16) and solving for ². Obviously, for n2s = x2 , large ²s implies large a ˜ independent of the boundedness of x, which, in turn, implies that large ²s carries information about the parameter error a ˜ even when x 6∈ L∞ . This indicates that ns may be used to normalize the effect of the possible unbounded signal x and is, therefore, referred to as the normalizing signal. Because of the similarity of ²s with the normalized estimation error defined in (4.3.5), we refer to ² in (4.3.13), (4.3.16) as the normalized estimation error too. Remark 4.3.1 The normalizing term ²n2s in (4.3.16) is similar to the nonlinear “damping” term used in the control of nonlinear systems [99]. It makes V˙ more negative by introducing the negative term −²2 n2s in the expression for V˙ and helps establish that ²ns ∈ L2 . Because a ˆ˙ is

168

CHAPTER 4. ON-LINE PARAMETER ESTIMATION

4 3.5 3 2.5

˜ a

2 1.5

α=10

1

α=2 0.5 0 0

α=0.5

5

10

15

20

25

30

35

40 sec

Figure 4.4 Effect of normalization on the convergence and performance of the adaptive law (4.3.15). p

bounded from above by ² n2s + 1 = ²m and ²m ∈ L2 , we can conclude that a ˆ˙ ∈ L2 , which is a desired property of the adaptive law. Note, however, that a ˆ˙ ∈ L2 does not imply that a ˆ˙ ∈ L∞ . In contrast to the example in Section 4.3.1, we have not been able to establish that a ˆ˙ ∈ L∞ . As we will show in Chapter 6 and 7, the L2 property of the derivative of the estimated parameters is sufficient to establish stability in the adaptive control case.

Simulations Let us simulate the effect of normalization on the convergence and performance of the adaptive law (4.3.15) when a = 0 is unknown, u = sin t, and am = 2. We use n2s = αx2 and consider different values of α ≥ 0. The simulation results are shown in Figure 4.4. It is clear that large values of α lead to a large normalizing signal that slows down the speed of convergence.

4.3. ADAPTIVE LAWS WITH NORMALIZATION

4.3.3

169

General Plant

Let us now consider the SISO plant x˙ = Ax + Bu, x(0) = x0 y = C >x

(4.3.18)

where x ∈ Rn and only y, u are available for measurement. Equation (4.3.18) may also be written as y = C > (sI − A)−1 Bu + C > (sI − A)−1 x0 or as y=

Z(s) C > adj(sI − A)x0 u+ R(s) R(s)

(4.3.19)

where Z(s), R(s) are in the form Z(s) = bn−1 sn−1 + bn−2 sn−2 + · · · + b1 s + b0 R(s) = sn + an−1 sn−1 + · · · + a1 s + a0 The constants ai , bi for i = 0, 1, . . . , n − 1 are the plant parameters. A convenient parameterization of the plant that allows us to extend the results of the previous sections to this general case is the one where the unknown parameters are separated from signals and expressed in the form of a linear equation. Several such parameterizations have already been explored and presented in Chapter 2. We summarize them here and refer to Chapter 2 for the details of their derivation. Let θ∗ = [bn−1 , bn−2 , . . . , b1 , b0 , an−1 , an−2 , . . . , a1 , a0 ]> be the vector with the unknown plant parameters. The vector θ∗ is of dimension 2n. If some of the coefficients of Z(s) are zero and known, i.e., Z(s) is of degree m < n − 1 where m is known, the dimension of θ∗ may be reduced. Following the results of Chapter 2, the plant (4.3.19) may take any one of the following parameterizations: z = θ∗> φ + η0

(4.3.20)

y = θλ∗> φ + η0

(4.3.21)

170

CHAPTER 4. ON-LINE PARAMETER ESTIMATION y = W (s)θλ∗> ψ + η0

(4.3.22)

where "

z = W1 (s)y,

φ = H(s)

u y

#

"

,

ψ = H1 (s)

u y

#

,

Λc t η0 = c> B0 x0 0e

θλ∗ = θ∗ − bλ W1 (s), H(s), H1 (s) are some known proper transfer function matrices with stable poles, bλ = [0, λ> ]> is a known vector, and Λc is a stable matrix which makes η0 to be an exponentially decaying to zero term that is due to nonzero initial conditions. The transfer function W (s) is a known strictly proper transfer function with relative degree 1, stable poles, and stable zeros. Instead of dealing with each parametric model separately, we consider the general model z = W (s)θ∗> ψ + η0

(4.3.23)

where W (s) is a proper transfer function with stable poles, z ∈ R1 , ψ ∈ R2n Λc t B x . Initially are signal vectors available for measurement and η0 = c> 0 0 0e we will assume that η0 = 0, i.e., z = W (s)θ∗> ψ

(4.3.24)

and use (4.3.24) to develop adaptive laws for estimating θ∗ on-line. The effect of η0 and, therefore, of the initial conditions will be treated in Section 4.3.7. Because θ∗ is a constant vector, going from form (4.3.23) to form (4.3.20) is trivial, i.e., rewrite (4.3.23) as z = θ∗> W (s)ψ + η0 and define φ = W (s)ψ. As illustrated in Chapter 2, the parametric model (4.3.24) may also be a parameterization of plants other than the LTI one given by (4.3.18). What is crucial about (4.3.24) is that the unknown vector θ∗ appears linearly in an equation where all other signals and parameters are known exactly. For this reason we will refer to (4.3.24) as the linear parametric model. In the literature, (3.4.24) has also been referred to as the linear regression model. In the following section we use different techniques to develop adaptive laws for estimating θ∗ on-line by assuming that W (s) is a known, proper transfer function with stable poles, and z, ψ are available for measurement.

4.3. ADAPTIVE LAWS WITH NORMALIZATION

4.3.4

171

SPR-Lyapunov Design Approach

This approach dominated the literature of continuous adaptive schemes [48, 149, 150, 153, 172, 178, 187]. It involves the development of a differential equation that relates the estimation or normalized estimation error with the parameter error through an SPR transfer function. Once in this form the KYP or the MKY Lemma is used to choose an appropriate Lyapunov function V whose time derivative V˙ is made nonpositive, i.e., V˙ ≤ 0 by properly choosing the differential equation of the adaptive law. The development of such an error SPR equation had been a challenging problem in the early days of adaptive control [48, 150, 153, 178]. The efforts in those days were concentrated on finding the appropriate transformation or generating the appropriate signals that allow the expression of the estimation/parameter error equation in the desired form. In this section we use the SPR-Lyapunov design approach to design adaptive laws for estimating θ∗ in the parametric model (4.3.24). The connection of the parametric model (4.3.24) with the adaptive control problem is discussed in later chapters. By treating parameter estimation independently of the control design, we manage to separate the complexity of the estimation part from that of the control part. We believe this approach simplifies the design and analysis of adaptive control schemes, to be discussed in later chapters, and helps clarify some of the earlier approaches that appear tricky and complicated to the nonspecialist. Let us start with the linear parametric model z = W (s)θ∗> ψ

(4.3.25)

Because θ∗ is a constant vector, we can rewrite (4.3.25) in the form z = W (s)L(s)θ∗> φ

(4.3.26)

where φ = L−1 (s)ψ and L(s) is chosen so that L−1 (s) is a proper stable transfer function and W (s)L(s) is a proper SPR transfer function. Remark 4.3.2 For some W (s) it is possible that no L(s) exists such that W (s)L(s) is proper and SPR. In such cases, (4.3.25) could be properly manipulated and put in the form of (4.3.26). For example, when

172

CHAPTER 4. ON-LINE PARAMETER ESTIMATION W (s) = s−1 s+2 , no L(s) can be found to make W (s)L(s) SPR. In this s+1 case, we write (4.3.25) as z¯ = (s+2)(s+3) θ∗> φ where φ = s−1 s+1 ψ and s+1 1 z¯ = s+3 z. The new W (s) in this case is W (s) = (s+2)(s+3) and a wide class of L(s) can be found so that W L is SPR.

The significance of the SPR property of W (s)L(s) is explained as we proceed with the design of the adaptive law. Let θ(t) be the estimate of θ∗ at time t. Then the estimate zˆ of z at time t is constructed as zˆ = W (s)L(s)θ> φ (4.3.27) As with the examples in the previous section, the estimation error ²1 is generated as ²1 = z − zˆ and the normalized estimation error as ² = z − zˆ − W (s)L(s)²n2s = ²1 − W (s)L(s)²n2s

(4.3.28)

where ns is the normalizing signal which we design to satisfy φ ∈ L∞ , m2 = 1 + n2s m

(A1)

Typical choices for ns that satisfy (A1) are n2s = φ> φ, n2s = φ> P φ for any P = P > > 0, etc. When φ ∈ L∞ , (A1) is satisfied with m = 1, i.e., ns = 0 in which case ² = ²1 . We examine the properties of ² by expressing (4.3.28) in terms of the 4 parameter error θ˜ = θ − θ∗ , i.e., substituting for z, zˆ in (4.3.28) we obtain ² = W L(−θ˜> φ − ²n2s )

(4.3.29)

For simplicity, let us assume that L(s) is chosen so that W L is strictly proper and consider the following state space representation of (4.3.29): e˙ = Ac e + Bc (−θ˜> φ − ²n2s ) ² = Cc> e

(4.3.30)

where Ac , Bc , and Cc are the matrices associated with a state space representation that has a transfer function W (s)L(s) = Cc> (sI − Ac )−1 Bc .

4.3. ADAPTIVE LAWS WITH NORMALIZATION

173

The error equation (4.3.30) relates ² with the parameter error θ˜ and is used to construct an appropriate Lyapunov type function for designing the adaptive law of θ. Before we proceed with such a design, let us examine (4.3.30) more closely by introducing the following remark. Remark 4.3.3 The normalized estimation error ² and the parameters Ac , Bc , and Cc in (4.3.30) can be calculated from (4.3.28) and the knowledge of W L, respectively. However, the state error e cannot be measured or generated because of the unknown input θ˜> φ. Let us now consider the following Lyapunov-like function for the differential equation (4.3.30): > ˜> −1 ˜ ˜ e) = e Pc e + θ Γ θ V (θ, 2 2

(4.3.31)

where Γ = Γ> > 0 is a constant matrix and Pc = Pc> > 0 satisfies the algebraic equations > Pc Ac + A> c Pc = −qq − νLc Pc Bc = Cc

(4.3.32)

for some vector q, matrix Lc = L> c > 0 and a small constant ν > 0. Equation (4.3.32) is guaranteed by the SPR property of W (s)L(s) = Cc> (sI −Ac )−1 Bc and the KYL Lemma if (Ac , Bc , Cc ) is minimal or the MKY Lemma if (Ac , Bc , Cc ) is nonminimal. Remark 4.3.4 Because the signal vector φ in (4.3.30) is arbitrary and could easily be the state of another differential equation, the function (4.3.31) is not guaranteed to be positive definite in a space that includes φ. Hence, V is a Lyapunov-like function. The time derivative V˙ along the solution of (4.3.30) is given by ˜ e) = − 1 e> qq > e − ν e> Lc e + e> Pc Bc [−θ˜> φ − ²n2 ] + θ˜> Γ−1 θ˜˙ (4.3.33) V˙ (θ, s 2 2 ˙ We now need to choose θ˜ = θ˙ as a function of signals that can be measured so that the indefinite terms in V˙ are canceled out. Because e is not available

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

for measurement, θ˙ cannot depend on e explicitly. Therefore, at first glance, it seems that the indefinite term −e> Pc Bc θ˜> φ = −θ˜> φe> Pc Bc cannot be ˙ cancelled because the choice θ˙ = θ˜ = Γφe> Pc Bc is not acceptable due to the presence of the unknown signal e. Here, however, is where the SPR property of W L becomes handy. We know from (4.3.32) that Pc Bc = Cc which implies that e> Pc Bc = e> Cc = ². Therefore, (4.3.33) can be written as ˜ e) = − 1 e> qq > e − ν e> Lc e − ²θ˜> φ − ²2 n2 + θ˜> Γ−1 θ˜˙ V˙ (θ, s 2 2

(4.3.34)

˙ The choice for θ˜ = θ˙ to make V˙ ≤ 0 is now obvious, i.e., for ˙ θ˙ = θ˜ = Γ²φ

(4.3.35)

we have

˜ e) = − 1 e> qq > e − ν e> Lc e − ²2 n2 ≤ 0 V˙ (θ, (4.3.36) s 2 2 which together with (4.3.31) implies that V, e, ², θ, θ˜ ∈ L∞ and that ˜ e(t)) = V∞ < ∞ lim V (θ(t),

t→∞

Furthermore, it follows from (4.3.36) that Z ∞ 0

²2 n2s dτ +

ν 2

Z ∞ 0

˜ e> Lc edτ ≤ V (θ(0), e(0)) − V∞

(4.3.37)

˜ Because λmin (Lc )|e|2 ≤ e> Lc e and V (θ(0), e(0)) is finite for any finite initial condition, (4.3.37) implies that ²ns , e ∈ L2 and therefore ² = Cc> e ∈ L2 . From the adaptive law (4.3.35), we have ˙ ≤ kΓk|²m| |θ|

|φ| m

where m2 = 1 + n2s . Since ²2 m2 = ²2 + ²2 n2s and ², ²ns ∈ L2 we have that ˙ ²m ∈ L2 , which together with |φ| ∈ L∞ implies that θ˙ = θ˜ ∈ L2 . m

We summarize the properties of (4.3.35) by the following theorem. Theorem 4.3.1 The adaptive law (4.3.35) guarantees that

4.3. ADAPTIVE LAWS WITH NORMALIZATION

−W (s)L(s) ¾

+ z + ? Σl − 6

-

175

n2s ¾ •

1

- ²

θ(0)

zˆ = θ>φ

× ¾

Z t?

θ

Γ

0

(·)dt ¾

²φ

× ¾

6

6

φ

φ

Figure 4.5 Block diagram for implementing adaptive algorithm (4.3.35) with normalized estimation error. (i) (ii)

θ, ² ∈ L∞ ², ²ns , θ˙ ∈ L2

independent of the boundedness properties of φ. Remark 4.3.5 Conditions (i) and (ii) of Theorem 4.3.1 specify the quality of estimation guaranteed by the adaptive law (4.3.35). In Chapters 6 and 7, we will combine (4.3.35) with appropriate control laws to form adaptive control schemes. The stability properties of these schemes depend on the properties (i) and (ii) of the adaptive law. Remark 4.3.6 The adaptive law (4.3.35) using the normalized estimation error generated by (4.3.28) can be implemented using the block diagram shown in Figure 4.5. When W (s)L(s) = 1, the normalized estimation error becomes ² = ²1 /(1 + n2s ) with ²1 = z − zˆ, which is the same normalization used in the gradient algorithm. This result can be obtained using simple block diagram transformation, as illustrated in Figure 4.6. Remark 4.3.7 The normalizing effect of the signal ns can be explained by setting e˙ = 0 in (4.3.30) and solving for the “quasi” steady-state

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

n2s z

− + ? Σl 6−

-

¾

-

1

²

z − zˆ -

1 1 + n2s

-²

zˆ Figure 4.6 Two equivalent block diagrams for generating the normalized estimation error when W (s)L(s) = 1. response ²ss of ², i.e., ²ss =

²1ss α(−θ˜> φ) = 2 1 + αns 1 + αn2s

(4.3.38)

where α = −Cc> A−1 c Bc is positive, i.e., α > 0, because of the SPR property of W L and ²1ss is the “quasi” steady state response of ²1 . Because of ns , ²ss cannot become unbounded as a result of a possibly unbounded signal φ. Large ²ss implies that θ˜ is large; therefore, large ˜ which is less affected by φ. ² carries information about θ, Remark 4.3.8 The normalizing signal ns may be chosen as n2s = φ> φ or as n2s = φ> P (t)φ where P (t) = P > (t) > 0 has continuous bounded elements. In general, if we set ns = 0 we cannot establish that θ˙ ∈ L2 which, as we show in later chapters, is a crucial property for establishing stability in the adaptive control case. In some special cases, we can afford to set ns = 0 and still establish that θ˙ ∈ L2 . For example, if φ ∈ L∞ or if θ, ² ∈ L∞ implies that φ ∈ L∞ , then it follows from (4.3.35) that ² ∈ L2 ⇒ θ˙ ∈ L2 . When n2s = 0, i.e., m = 1, we refer to (4.3.35) as the unnormalized adaptive law. In this case ² = ²1 leading to the type of adaptive laws considered in Section 4.2. In later chapters, we show how to use both the normalized and unnormalized adaptive laws in the design of adaptive control schemes. Another desired property of the adaptive law (4.3.35) is the convergence

4.3. ADAPTIVE LAWS WITH NORMALIZATION

177

of θ(t) to the unknown vector θ∗ . Such a property is achieved for a special class of vector signals φ described by the following definition: Definition 4.3.1 (Persistence of Excitation (PE)) A piecewise continuous signal vector φ : R+ 7→ Rn is PE in Rn with a level of excitation α0 > 0 if there exist constants α1 , T0 > 0 such that 1 α1 I ≥ T0

Z t+T0 t

φ(τ )φ> (τ )dτ ≥ α0 I,

∀t ≥ 0

(4.3.39)

Although the matrix φ(τ )φ> (τ ) is singular for each τ , (4.3.39) requires that φ(t) varies in such a way with time that the integral of the matrix φ(τ )φ> (τ ) is uniformly positive definite over any time interval [t, t + T0 ]. If we express (4.3.39) in the scalar form, i.e., 1 α1 ≥ T0

Z t+T0 t

(q > φ(τ ))2 dτ ≥ α0 ,

∀t ≥ 0

(4.3.40)

where q is any constant vector in Rn with |q| = 1, then the condition can be interpreted as a condition on the energy of φ in all directions. The properties of PE signals as well as various other equivalent definitions and interpretations are given in the literature [1, 12, 22, 24, 52, 75, 127, 141, 171, 172, 201, 242]. Corollary 4.3.1 If ns , φ, φ˙ ∈ L∞ and φ is PE, then (4.3.35) guarantees that θ(t) → θ∗ exponentially fast. The proof of Corollary 4.3.1 is long and is given in Section 4.8. Corollary 4.3.1 is important in the case where parameter convergence is one of the primary objectives of the adaptive system. We use Corollary 4.3.1 in Chapter 5 to establish parameter convergence in parameter identifiers and adaptive observers for stable plants. The condition that φ˙ appears only in the case of the adaptive laws based on the SPR-Lyapunov approach with W (s)L(s) strictly proper. It is a condition in Lemma 4.8.3 (iii) that is used in the proof of Corollary 4.3.1 in Section 4.8. The results of Theorem 4.3.1 are also valid when W (s)L(s) is biproper (see Problem 4.4). In fact if W (s) is minimum phase, one may choose L(s) =

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

W −1 (s) leading to W (s)L(s) = 1. For W L = 1, (4.3.28), (4.3.29) become ²=

θ˜> φ z − zˆ = − m2 m2

where m2 = 1 + n2s . In this case we do not need to employ the KYP or MKY Lemma because the Lyapunov-like function ˜> −1 ˜ ˜ =θ Γ θ V (θ) 2 leads to V˙ = −²2 m2 by choosing θ˙ = Γ²φ

(4.3.41)

The same adaptive law as (4.3.41) can be developed by using the gradient method to minimize a certain cost function of ² with respect to θ. We discuss this method in the next section. Example 4.3.1 Let us consider the following signal y = A sin(ωt + ϕ) that is broadcasted with a known frequency ω but an unknown phase ϕ and unknown amplitude A. The signal y is observed through a device with transfer function W (s) that is designed to attenuate any possible noise present in the measurements of y, i.e., z = W (s)y = W (s)A sin(ωt + ϕ) (4.3.42) For simplicity let us assume that W (s) is an SPR transfer function. Our objective is to use the knowledge of the frequency ω and the measurements of z to estimate A, ϕ. Because A, ϕ may assume different constant values at different times we would like to use the results of this section and generate an on-line estimation scheme that provides continuous estimates for A, ϕ. The first step in our approach is to transform (4.3.42) in the form of the linear parametric model (4.3.24). This is done by using the identity A sin(ωt + ϕ) = A1 sin ωt + A2 cos ωt where A1 = A cos ϕ,

A2 = A sin ϕ

to express (4.3.42) in the form z = W (s)θ∗> φ

(4.3.43)

4.3. ADAPTIVE LAWS WITH NORMALIZATION

179

where θ∗ = [A1 , A2 ]> and φ = [sin ωt, cos ωt]> . From the estimate of θ∗ , i.e., A1 , A2 , we can calculate the estimate of A, ϕ by using the relationship (4.3.43). Using the results of this section the estimate θ(t) of θ∗ at each time t is given by θ˙ = Γ²φ ² = z − zˆ − W (s)²n2s ,

zˆ = W (s)θ> φ,

n2s = αφ> φ

where θ = [Aˆ1 , Aˆ2 ]> and Aˆ1 , Aˆ2 is the estimate of A1 , A2 , respectively. Since φ ∈ L∞ , the normalizing signal may be taken to be equal to zero, i.e., α = 0. The adaptive gain Γ may be chosen as Γ = diag(γ) for some γ > 0, leading to ˙ Aˆ1 = γ² sin ωt,

˙ Aˆ2 = γ² cos ωt

(4.3.44)

˙ ˙ The above adaptive law guarantees that Aˆ1 , Aˆ2 , ² ∈ L∞ and ², Aˆ1 , Aˆ2 ∈ L2 . Since ˙ˆ ˙ˆ φ ∈ L∞ we also have that A1 , A2 ∈ L∞ . As we mentioned earlier the convergence of Aˆ1 , Aˆ2 to A1 , A2 respectively is guaranteed provided φ is PE. We check the PE property of φ by using (4.3.39). We have 1 T0

Z

t+T0

φ(τ )φ> (τ )dτ =

t

1 S(t, T0 ) T0

where 

T0 sin 2ω(t + T0 ) − sin 2ωt − 4  S(t, T0 ) =  2 cos 2ω(t + T 4ω 0 ) − cos 2ωt − 4ω For T0 =

2π ω ,

 cos 2ω(t + T0 ) − cos 2ωt −  4ω T0 sin 2ω(t + T0 ) − sin 2ωt  + 2 4ω

we have 1 T0

Z

t+T0 t

· φ(τ )φ> (τ )dτ =

π ω

0

0

¸

π ω

Hence, the PE condition (4.3.39) is satisfied with T0 = 2π/ω, 0 < α0 ≤ π/ω, α1 ≥ π ˆ ˆ ω ; therefore, φ is PE, which implies that A1 , A2 converge to A1 , A2 exponentially fast. ˆ ϕˆ of A, ϕ, respectively, is calculated as follows: Using (4.3.43), the estimate A, Ã ! q Aˆ1 (t) −1 2 2 ˆ ˆ ˆ A(t) = A1 (t) + A2 (t), ϕˆ = cos (4.3.45) ˆ A(t) ˆ The calculation of ϕˆ at each time t is possible provided A(t) 6= 0. This implies that Aˆ1 , Aˆ2 should not go through zero at the same time, which is something that cannot be guaranteed by the adaptive law (4.3.44). We know, however, that Aˆ1 , Aˆ2

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

converge to A1 , A2 exponentially fast, and A1 , A2 cannot be both equal to zero (otherwise y ≡ 0 ∀t ≥ 0). Hence, after some finite time T , Aˆ1 , Aˆ2 will be close ˆ ) 6= 0 to imply A(t) ˆ 6= 0 ∀t ≥ T . enough to A1 , A2 for A(T ˆ ϕˆ Because Aˆ1 , Aˆ2 → A1 , A2 exponentially fast, it follows from (4.3.45) that A, converge to A, ϕ exponentially fast. 2 Let us simulate the above estimation scheme when W (s) = s+2 , ω = 2 rad/sec and the unknown A, ϕ are taken as A = 10, ϕ = 16◦ = 0.279 rad for 0 ≤ t ≤ 20 sec and A = 7, ϕ = 25◦ = 0.463 rad for t > 20 sec for simulation purposes. The results are shown in Figure 4.7, where γ = 1 is used. 5 Example 4.3.2 Consider the following plant: y=

b1 s + b0 u s2 + 3s + 2

where b1 , b0 are the only unknown parameters to be estimated. We rewrite the plant in the form of the parametric model (4.3.24) by first expressing it as y=

1 θ∗> ψ (s + 1)(s + 2)

(4.3.46)

1 where θ∗= [b1 , b0 ]>, ψ = [u, ˙ u]> . We then choose L(s) = s + 2 so that W (s)L(s) = s+1 is SPR and rewrite (4.3.46) as

y= h where φ =

1 ∗> θ φ s+1

(4.3.47)

i>

s 1 s+2 u, s+2 u

can be generated by filtering u. Because (4.3.47) is in the form of parametric model (4.3.24), we can apply the results of this section to obtain the adaptive law θ˙ = Γ²φ ²=y−

1 (θ> φ + ²n2s ), s+1

ns = αφ> φ

where α > 0 and θ = [ˆb1 , ˆb0 ]> is the on-line estimate of θ∗ . This example illustrates that the dimensionality of θ, φ may be reduced if some of the plant parameters are known. 5

4.3.5

Gradient Method

Some of the earlier approaches to adaptive control in the early 1960s [20, 34, 96, 104, 115, 123, 175, 220] involved the use of simple optimization techniques

4.3. ADAPTIVE LAWS WITH NORMALIZATION

181

2

15 Aˆ A

10

ˆϕ

1.5

ϕ

1 5

0

0.5

0

10

20 (a)

30

0 0

40

10

20 (b)

sec

30

40 sec

20 Aˆ ( t ) sin ( 2 t + ˆϕ ( t ) ) A sin ( 2 t + ϕ )

10 0 -10 -20

0

5

10

15

20 (c)

25

30

35

40

sec

Figure 4.7 Simulation results for Example 4.3.1.

such as the gradient or steepest descent method to minimize a certain performance cost with respect to some adjustable parameters. These approaches led to the development of a wide class of adaptive algorithms that had found wide applications in industry. Despite their success in applications, the schemes of the 1960s lost their popularity because of the lack of stability in a global sense. As a result, starting from the late 1960s and early 1970s, the schemes of the 1960s have been replaced by new schemes that are based on Lyapunov theory. The gradient method, however, as a tool for designing adaptive laws retained its popularity and has been widely used in discretetime [73] and, to a less extent, continuous-time adaptive systems. In contrast to the schemes of the 1960s, the schemes of the 1970s and 1980s that are based on gradient methods are shown to have global stability properties.

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

What made the difference with the newer schemes were new formulations of the parameter estimation problem and the selection of different cost functions for minimization. In this section, we use the gradient method and two different cost functions to develop adaptive laws for estimating θ∗ in the parametric model z = W (s)θ∗> ψ (4.3.24) The use of the gradient method involves the development of an algebraic estimation error equation that motivates the selection of an appropriate cost function J(θ) that is convex over the space of θ(t), the estimate of θ∗ at time t, for each time t. The function J(θ) is then minimized with respect to θ for each time t by using the gradient method described in Appendix B. The algebraic error equation is developed as follows: Because θ∗ is constant, the parametric model (4.3.24) can be written in the form z = θ∗> φ

(4.3.48)

where φ = W (s)ψ. The parametric model (4.3.48) has been the most popular one in discrete time adaptive control. At each time t, (4.3.48) is an algebraic equation where the unknown θ∗ appears linearly. Because of the simplicity of (4.3.48), a wide class of recursive adaptive laws may be developed. Using (4.3.48) the estimate zˆ of z at time t is generated as zˆ = θ> φ where θ(t) is the estimate of θ∗ at time t. The normalized estimation error ² is then constructed as ²=

z − zˆ z − θ> φ = m2 m2

(4.3.49)

where m2 = 1 + n2s and ns is the normalizing signal designed so that φ ∈ L∞ m

(A1)

As in Section 4.3.4, typical choices for ns are n2s = φ> φ, n2s = φ> P φ for P = P > > 0, etc.

4.3. ADAPTIVE LAWS WITH NORMALIZATION

183

For analysis purposes we express ² as a function of the parameter error θ˜ = θ − θ∗ , i.e., substituting for z in (4.3.49) we obtain 4

²=−

θ˜> φ m2

(4.3.50)

φ Clearly the signal ²m = −θ˜> m is a reasonable measure of the parameter ˜ error θ because for any piecewise continuous signal vector φ (not necessarily ˜ Several adaptive laws for θ can be bounded), large ²m implies large θ. generated by using the gradient method to minimize a wide class of cost functions of ² with respect to θ. In this section we concentrate on two different cost functions that attracted considerable interest in the adaptive control community.

Instantaneous Cost Function Let us consider the simple quadratic cost function J(θ) =

(z − θ> φ)2 ² 2 m2 = 2 2m2

(4.3.51)

motivated from (4.3.49), (4.3.50), that we like to minimize with respect to θ. Because of the property (A1) of m, J(θ) is convex over the space of θ at each time t; therefore, the minimization problem is well posed. Applying the gradient method, the minimizing trajectory θ(t) is generated by the differential equation θ˙ = −Γ∇J(θ) where Γ = Γ> > 0 is a scaling matrix that we refer to as the adaptive gain. From (4.3.51) we have ∇J(θ) = −

(z − θ> φ)φ = −²φ m2

and, therefore, the adaptive law for generating θ(t) is given by θ˙ = Γ²φ We refer to (4.3.52) as the gradient algorithm.

(4.3.52)

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

Remark 4.3.9 The adaptive law (4.3.52) has the same form as (4.3.35) developed using the Lyapunov design approach. As shown in Section 4.3.4, (4.3.52) follows directly from the Lyapunov design method by taking L(s) = W −1 (s). Remark 4.3.10 The convexity of J(θ) (as explained in Appendix B) guarantees the existence of a single global minimum defined by ∇J(θ) = 0. >φ Solving ∇J(θ) = −²φ = − z−θ φ = 0, i.e., φz = φφ> θ, for θ will give m2 us the nonrecursive gradient algorithm θ(t) = (φφ> )−1 φz provided that φφ> is nonsingular. For φ ∈ Rn×1 and n > 1, φφ> is always singular, the following nonrecursive algorithm based on N data points could be used: θ(t) =

ÃN X i=1

>

!−1 N X

φ(ti )φ (ti )

φ(ti )z(ti )

i=1

where ti ≤ t, i = 1, . . . , N are the points in time where the measurements of φ and z are taken. Remark 4.3.11 The minimum of J(θ) corresponds to ² = 0, which implies θ˙ = 0 and the end of adaptation. The proof that θ(t) will converge to a trajectory that corresponds to ² being small in some sense is not directly guaranteed by the gradient method. A Lyapunov type of analysis is used to establish such a result as shown in the proof of Theorem 4.3.2 that follows.

Theorem 4.3.2 The adaptive law (4.3.52) guarantees that (i) (ii)

², ²ns , θ, θ˙ ∈ L∞ ², ²ns , θ˙ ∈ L2

independent of the boundedness of the signal vector φ and (iii) if ns , φ ∈ L∞ and φ is PE, then θ(t) converges exponentially to θ∗

4.3. ADAPTIVE LAWS WITH NORMALIZATION

185

˙ Proof Because θ∗ is constant, θ˜ = θ˙ and from (4.3.52) we have ˙ θ˜ = Γ²φ

(4.3.53)

We choose the Lyapunov-like function ˜ = V (θ)

θ˜> Γ−1 θ˜ 2

Then along the solution of (4.3.53), we have V˙ = θ˜> φ² = −²2 m2 ≤ 0

(4.3.54)

where the second equality is obtained by substituting θ˜> φ = −²m2 from (4.3.50). Hence, V, θ˜ ∈ L∞ , which, together with (4.3.50), implies that ², ²m ∈ L∞ . In addition, we establish from the properties of V, V˙ , by applying the same argument as in the previous sections, that ²m ∈ L2 , which implies that ², ²ns ∈ L2 . Now from (4.3.53) we have ˜˙ = |θ| ˙ ≤ kΓk|²m| |φ| |θ| (4.3.55) m T T which together with |φ| L∞ implies that θ˙ ∈ L2 L∞ and m ∈ L∞ and ²m ∈ L2 the proof for (i) and (ii) is complete. The proof for (iii) is long and more complicated and is given in Section 4.8. 2

˜ ≥ 0 and V˙ ≤ 0 of the Lyapunov-like Remark 4.3.12 The property V (θ) ˜ function implies that limt→∞ V (θ(t)) = V∞ . This, however, does not imply that V˙ (t) goes to zero as t → ∞. Consequently, we cannot conclude that ² or ²m go to zero as t → ∞, i.e., that the steepest descent reaches the global minimum that corresponds to ∇J(θ) = −²φ = 0. d ˙ If however, φ/m, m/m ˙ ∈ L∞ , we can establish that dt (²m) ∈ L∞ , which, together with ²m ∈ L2 , implies that ²(t)m(t) → 0 as t → ∞. Because m2 = 1 + n2s we have ²(t) → 0 as t → ∞ and from (4.3.55) ˙ → 0 as t → ∞. Now |∇J(θ)| ≤ |²φ| ≤ |²m| |φ| , which implies that θ(t) m that |∇J(θ(t))| → 0 as t → ∞, i.e., θ(t) converges to a trajectory that corresponds to a global minimum of J(θ) asymptotically with time φ˙ m provided m , m˙ ∈ L∞ . Remark 4.3.13 Even though the form of the gradient algorithm (4.3.52) is the same as that of the adaptive law (4.3.35) based on the SPRLyapunov design approach, their properties are different. For example,

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION (4.3.52) guarantees that θ˙ ∈ L∞ , whereas such property has not been shown for (4.3.35).

The speed of convergence of the estimated parameters to their true values, when ns , φ ∈ L∞ and φ is PE, is characterized in the proof of Theorem 4.3.2 (iii) in Section 4.8. It is shown that ˜ ≤ γ n θ˜> (0)Γ−1 θ(0) ˜ θ˜> (t)Γ−1 θ(t) where 0 ≤ t ≤ nT0 , n is an integer and γ = 1 − γ1 , γ1 =

2α0 T0 λmin (Γ) 2m0 + β 4 T02 λ2max (Γ)

where α0 is the level of excitation of φ, T0 > 0 is the size of the time interval in the PE definition of φ, m0 = supt≥0 m2 (t) and β = supt≥0 |φ(t)|. We established that 0 < γ < 1. The smaller the γ, i.e., the larger the γ1 , the faster the parameter error converges to zero. The constants α0 , T0 , β and possibly m0 are all interdependent because they all depend on φ(t). It is, therefore, not very clear how to choose φ(t), if we can, to increase the size of γ1 . Integral Cost Function A cost function that attracted some interest in the literature of adaptive systems [108] is the integral cost function J(θ) =

1 2

Z t 0

e−β(t−τ ) ²2 (t, τ )m2 (τ )dτ

(4.3.56)

where β > 0 is a design constant and ²(t, τ ) =

z(τ ) − θ> (t)φ(τ ) , m2 (τ )

²(t, t) = ²

(4.3.57)

is the normalized estimation error at time τ based on the estimate θ(t) of θ∗ at time t ≥ τ . The design constant β acts as a forgetting factor, i.e., as time t increases the effect of the old data at time τ < t is discarded exponentially. The parameter θ(t) is to be chosen at each time t to minimize the integral square of the error on all past data that are discounted exponentially.

4.3. ADAPTIVE LAWS WITH NORMALIZATION

187

Using (4.3.57), we express (4.3.56) in terms of the parameter θ, i.e., J(θ) =

1 2

Z t 0

e−β(t−τ )

(z(τ ) − θ> (t)φ(τ ))2 dτ m2 (τ )

(4.3.58)

Clearly, J(θ) is convex over the space of θ for each time t and the application of the gradient method for minimizing J(θ) w.r.t. θ yields θ˙ = −Γ∇J = Γ

Z t 0

e−β(t−τ )

(z(τ ) − θ> (t)φ(τ )) φ(τ )dτ m2 (τ )

(4.3.59)

where Γ = Γ> > 0 is a scaling matrix that we refer to as the adaptive gain. Equation (4.3.59) is implemented as θ˙ = −Γ(R(t)θ + Q(t)) φφ> R˙ = −βR + 2 , R(0) = 0 m zφ Q˙ = −βQ − 2 , Q(0) = 0 m

(4.3.60)

where R ∈ Rn×n , Q ∈ Rn×1 . We refer to (4.3.59) or (4.3.60) as the integral adaptive law. Its form is different from that of the previous adaptive laws we developed. The properties of (4.3.60) are also different and are given by the following theorem. Theorem 4.3.3 The integral adaptive law (4.3.60) guarantees that (i) (ii) (iii) (iv)

², ²ns , θ, θ˙ ∈ L∞ ², ²ns , θ˙ ∈ L2 ˙ limt→∞ |θ(t)| =0 if ns , φ ∈ L∞ and φ is PE then θ(t) converges exponentially to θ∗ . Furthermore, for Γ = γI the rate of convergence can be made arbitrarily large by increasing the value of the adaptive gain γ.

φ Proof Because m ∈ L∞ , it follows that R, Q ∈ L∞ and, therefore, the differential equation for θ behaves as a linear time-varying differential equation with a bounded input. Substituting for z = φ> θ∗ in the differential equation for Q we verify that Z t φ(τ )φ> (τ ) dτ θ∗ = −R(t)θ∗ (4.3.61) Q(t) = − e−β(t−τ ) m2 0

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

and, therefore,

˙ θ˙ = θ˜ = −ΓR(t)θ˜

(4.3.62)

We analyze (4.3.62) by using the Lyapunov-like function ˜> −1 ˜ ˜ =θ Γ θ V (θ) 2

(4.3.63)

whose time derivative along the solution of (4.3.62) is given by V˙ = −θ˜> R(t)θ˜

(4.3.64)

˜ θ ∈ L∞ , Because R(t) = R> (t) ≥ 0 ∀t ≥ 0 it follows that V˙ ≤ 0; therefore, V, θ, > 1 1 ˜ ˜ 2 = |R 2 θ| ˜ ∈ L2 . From ² = − θ 2φ and θ, ˜ φ ∈ L∞ we conclude that ², ²m (θ˜> Rθ) m m and, therefore, ²ns ∈ L∞ . From (4.3.62) we have ˙ ≤ kΓR >2 k|R 12 θ| ˜ |θ| (4.3.65) T T 1 ˜ ∈ L∞ L2 imply that θ˙ ∈ L∞ L2 . Since which together with R ∈ L∞ and |R 2 θ| ˙˜ ˙ ¨ ˙ θ, R ∈ L∞ , it follows from (4.3.62) that θ˜ ∈ L∞ , which, together with θ˜ ∈ L2 , ˙ ˜ implies limt→∞ |θ(t)| = limt→∞ |ΓR(t)θ(t)| = 0. To show that ²m ∈ L2 we proceed as follows. We have d ˜> ˜ θ Rθ = ²2 m2 − 2θ˜> RΓRθ˜ − β θ˜> Rθ˜ dt Therefore, Z

t

Z ²2 m2 dτ = θ˜> Rθ˜ + 2

0

t

Z ˜ +β θ˜> RΓRθdτ

0

t

˜ θ˜> Rθdτ

0

˜ ˜ ∈ L2 it follows that Because limt→∞ [θ˜> (t)R(t)θ(t)] = 0 and |R θ| 1 2

Z

t

lim

t→∞

0

Z ²2 m2 dτ =

∞

²2 m2 dτ < ∞

0

i.e., ²m ∈ L2 . Hence, the proof for (i) to (iii) is complete. The proof for (iv) is given in Section 4.8. 2

Remark 4.3.14 In contrast to the adaptive law based on the instantaneous ˙ = −Γ∇J(θ(t)) → 0 cost, the integral adaptive law guarantees that θ(t) as t → ∞ without any additional conditions on the signal vector φ

4.3. ADAPTIVE LAWS WITH NORMALIZATION

189

and m. In this case, θ(t) converges to a trajectory that minimizes the integral cost asymptotically with time. As we demonstrated in ˙ Chapter 3 using simple examples, the convergence of θ(t) to a zero vector does not imply that θ(t) converges to a constant vector. In the proof of Theorem 4.3.3 (iv) given in Section 4.8, we have established that when ns , φ ∈ L∞ and φ is PE, the parameter error θ˜ satisfies s

˜ |θ(t)| ≤

−α λmax (Γ) ˜ |θ(T0 )|e 2 (t−T0 ) , ∀t ≥ T0 λmin (Γ)

where 0

0

α = 2β1 e−βT0 λmin (Γ), β1 = α0 T0 α0 , α0 = sup t

1 m2 (t)

and α0 , T0 are constants in the definition of the PE property of φ, i.e., α0 > 0 is the level of excitation and T0 > 0 is the length of the time interval. The ˜ size of the constant α > 0 indicates the speed with which |θ(t)| is guaranteed to converge to zero. The larger the level of excitation α0 , the larger the α is. A large normalizing signal decreases the value of α and may have a negative effect on the speed of convergence. If Γ is chosen as Γ = γI, then it becomes ˜ clear that a larger γ guarantees a faster convergence of |θ(t)| to zero. Example 4.3.3 Let us consider the same problem as in Example 4.3.1. We consider the equation z = W (s)A sin(ωt + ϕ) = W (s)θ∗> φ (4.3.66) where θ∗ = [A1 , A2 ]> , φ = [sin ωt, cos ωt]> , A1 = A cos ϕ, A2 = A sin ϕ. We need to estimate A, ϕ using the knowledge of φ, ω, W (s) and the measurement of z. We first express (4.3.66) in the form of the linear parametric model (4.3.48) by filtering φ with W (s), i.e., z = θ∗> φ0 ,

φ0 = W (s)φ

(4.3.67)

and then obtain the adaptive law for estimating θ∗ by applying the results of this section. The gradient algorithm based on the instantaneous cost is given by z − θ > φ0 θ˙ = Γ²φ0 , ² = , m2

m2 = 1 + αφ> 0 φ0

(4.3.68)

where θ = [Aˆ1 , Aˆ2 ]> is the estimate of θ∗ and α ≥ 0. Because φ ∈ L∞ and W (s) has stable poles, φ0 ∈ L∞ and α can be taken to be equal to zero.

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

2

15 Aˆ A

10

ˆϕ

1.5

ϕ

1 5

0

0.5

0

10

20 (a)

30

0 0

40

10

20 (b)

sec

30

40 sec

20 Aˆ ( t ) sin ( 2 t + ˆϕ ( t ) ) A sin ( 2 t + ϕ )

10 0 -10 -20

0

5

10

15

20 (c)

25

30

35

40

sec

Figure 4.8 Simulation results for Example 4.3.3: Performance of the gradient adaptive law (4.3.68) based on the instantaneous cost. The gradient algorithm based on the integral cost is given by θ˙

=

R˙

=

Q˙ =

−Γ(R(t)θ + Q), θ(0) = θ0 φ0 φ> 0 −βR + , R(0) = 0 m2 zφ0 −βQ − 2 , Q(0) = 0 m

(4.3.69)

ˆ ϕˆ of the unknown constants A, ϕ is where R ∈ R2×2 , Q ∈ R2×1 . The estimate A, ˆ ˆ calculated from the estimates A1 , A2 in the same way as in Example 4.3.1. We can establish, as shown in Example 4.3.1, that φ0 satisfies the PE conditions; therefore, both adaptive laws (4.3.68) and (4.3.69) guarantee that θ converges to θ∗ exponentially fast. Let us now simulate (4.3.68) and (4.3.69). We choose Γ = diag(γ) with γ = 10 for both algorithms, and β = 0.1 for (4.3.69). We also use ω = 2 rad/sec and

4.3. ADAPTIVE LAWS WITH NORMALIZATION

191

2

15 Aˆ A

10

ˆϕ

1.5

ϕ

1 5

0

0.5

0

10

20 (a)

30

0 0

40

10

20 (b)

sec

30

40 sec

20 Aˆ ( t ) sin ( 2 t + ˆϕ ( t ) ) A sin ( 2 t + ϕ )

10 0 -10 -20

0

5

10

15

20 (c)

25

30

35

40

sec

Figure 4.9 Simulation results for Example 4.3.3: Performance of the gradient adaptive law (4.3.69) based on the integral cost. 2 W (s) = s+2 . Figure 4.8 shows the performance of the adaptive law (4.3.68) based on the instantaneous cost, and Figure 4.9 shows that of (4.3.69) based on the integral cost. 5

Remark 4.3.15 This example illustrates that for the same estimation problem the gradient method leads to adaptive laws that require more integrators than those required by adaptive laws based on the SPRLyapunov design approach. Furthermore, the gradient algorithm based on the integral cost is far more complicated than that based on the instantaneous cost. This complexity is traded off by the better convergence properties of the integral algorithm described in Theorem 4.3.3.

192

4.3.6

CHAPTER 4. ON-LINE PARAMETER ESTIMATION

Least-Squares

The least-squares is an old method dating back to Gauss in the eighteenth century where he used it to determine the orbit of planets. The basic idea behind the least-squares is fitting a mathematical model to a sequence of observed data by minimizing the sum of the squares of the difference between the observed and computed data. In doing so, any noise or inaccuracies in the observed data are expected to have less effect on the accuracy of the mathematical model. The method of least-squares has been widely used in parameter estimation both in a recursive and nonrecursive form mainly for discrete-time systems [15, 52, 73, 80, 127, 144]. The method is simple to apply and analyze in the case where the unknown parameters appear in a linear form, such as in the linear parametric model z = θ∗> φ

(4.3.70)

Before embarking on the use of least-squares to estimate θ∗ in (4.3.70), let us illustrate its use and properties by considering the simple scalar plant y = θ∗ u + dn

(4.3.71)

where dn is a noise disturbance; y, u ∈ R+ and u ∈ L∞ . We examine the following estimation problem: Given the measurements of y(τ ), u(τ ) for 0 ≤ τ < t, find a “good” estimate θ(t) of θ∗ at time t. One possible solution is to calculate θ(t) from θ(t) =

y(τ ) dn (τ ) = θ∗ + u(τ ) u(τ )

for some τ < t for which u(τ ) 6= 0. Because of the noise disturbance, however, such an estimate may be far off from θ∗ . A more natural approach is to generate θ by minimizing the cost function J(θ) =

1 2

Z t 0

(y(τ ) − θ(t)u(τ ))2 dτ

(4.3.72)

with respect to θ at any given time t. The cost J(θ) penalizes all the past errors from τ = 0 to t that are due to θ(t) = 6 θ∗ . Because J(θ) is a convex

4.3. ADAPTIVE LAWS WITH NORMALIZATION

193

function over R1 at each time t, its minimum satisfies ∇J(θ) = −

Z t 0

y(τ )u(τ )dτ + θ(t)

Z t 0

u2 (τ )dτ = 0

(4.3.73)

y(τ )u(τ )dτ

(4.3.74)

for any given time t, which gives θ(t) =

µZ t 0

2

u (τ )dτ

¶−1 Z t 0

provided of course the inverse exists. This is the celebrated least-squares estimate. The least-squares method considers all past data in an effort to provide a good estimate for θ∗ in the presence of noise dn . For example, when u(t) = 1 ∀t ≥ 0 and dn has a zero average value, we have 1 lim θ(t) = lim t→∞ t→∞ t

Z t 0

1 y(τ )u(τ )dτ = θ + lim t→∞ t ∗

Z t 0

dn (τ )dτ = θ∗

i.e., θ(t) converges to the exact parameter value despite the presence of the noise disturbance dn . Let us now extend this problem to the linear model (4.3.70). As in Section 4.3.5, the estimate zˆ of z and the normalized estimation error are generated as zˆ = θ> φ,

²=

z − zˆ z − θ> φ = 2 m m2

(4.3.49)

where m2 = 1 + n2s , θ(t) is the estimate of θ∗ at time t, and m satisfies φ/m ∈ L∞ . We consider the following cost function Z t

1 [z(τ ) − θ> (t)φ(τ )]2 dτ + e−βt (θ − θ0 )> Q0 (θ − θ0 ) 2 m (τ ) 2 0 (4.3.75) > where Q0 = Q0 > 0, β ≥ 0, θ0 = θ(0), which is a generalization of (4.3.72) to include discounting of past data and a penalty on the initial estimate θ0 of θ∗ . The cost (4.3.75), apart from the additional term that penalizes the initial parameter error, is identical to the integral cost (4.3.58) considered in Section 4.3.5. The method, however, for developing the estimate θ(t) for θ∗ is different. Because z/m, φ/m ∈ L∞ , J(θ) is a convex function of θ over Rn at each time t. Hence, any local minimum is also global and satisfies 1 J(θ) = 2

e−β(t−τ )

∇J(θ(t)) = 0,

∀t ≥ 0

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

i.e., ∇J(θ) = e

−βt

Q0 (θ(t) − θ0 ) −

Z t 0

e−β(t−τ )

z(τ ) − θ> (t)φ(τ ) φ(τ )dτ = 0 m2 (τ )

which yields the so-called nonrecursive least-squares algorithm ·

θ(t) = P (t) e−βt Q0 θ0 + where

"

P (t) = e−βt Q0 +

Z t 0

Z t 0

e−β(t−τ )

z(τ )φ(τ ) dτ m2 (τ )

φ(τ )φ> (τ ) e−β(t−τ ) dτ m2 (τ )

¸

(4.3.76)

#−1

(4.3.77)

> Because Q0 = Q> 0 > 0 and φφ is positive semidefinite, P (t) exists at each time t. Using the identity

d d P P −1 = P˙ P −1 + P P −1 = 0 dt dt we can show that P satisfies the differential equation >

φφ P˙ = βP − P 2 P, m

P (0) = P0 = Q−1 0

(4.3.78)

Therefore, the calculation of the inverse in (4.3.77) is avoided by generating P as the solution of the differential equation (4.3.78). Similarly, differentiating θ(t) w.r.t. t and using (4.3.78) and ²m2 = z − θ> φ, we obtain θ˙ = P ²φ

(4.3.79)

We refer to (4.3.79) and (4.3.78) as the continuous-time recursive leastsquares algorithm with forgetting factor. The stability properties of the least-squares algorithm depend on the value of the forgetting factor β as discussed below. Pure Least-Squares In the identification literature, (4.3.79) and (4.3.78) with β = 0 is referred to as the “pure” least-squares algorithm and has a very similar form as the Kalman filter. For this reason, the matrix P is usually called the covariance matrix.

4.3. ADAPTIVE LAWS WITH NORMALIZATION

195

Setting β = 0, (4.3.78), (4.3.79) become θ˙ = P ²φ P φφ> P , P˙ = − m2

P (0) = P0

(4.3.80)

In terms of the P −1 we have d −1 φφ> P = dt m2 −1

which implies that d(Pdt ) ≥ 0, and, therefore, P −1 may grow without bound. In the matrix case, this means that P may become arbitrarily small and slow down adaptation in some directions. This is the so-called covariance wind-up problem that constitutes one of the main drawbacks of the pure least-squares algorithm. Despite its deficiency, the pure least-squares algorithm has the unique property of guaranteeing parameter convergence to constant values as described by the following theorem: Theorem 4.3.4 The pure least-squares algorithm (4.3.80) guarantees that (i) (ii) (iii) (iv)

˙ P ∈ L∞ . ², ²ns , θ, θ, ², ²ns , θ˙ ∈ L2 . ¯ where θ¯ is a constant vector. limt→∞ θ(t) = θ, If ns , φ ∈ L∞ and φ is PE, then θ(t) converges to θ∗ as t → ∞.

Proof From (4.3.80) we have that P˙ ≤ 0, i.e., P (t) ≤ P0 . Because P (t) is nonincreasing and bounded from below (i.e., P (t) = P > (t) ≥ 0, ∀t ≥ 0) it has a limit, i.e., lim P (t) = P¯ t→∞

¯>

where P¯ = P

≥ 0 is a constant matrix. Let us now consider d −1 ˜ φφ> θ˜ ˙ (P θ) = −P −1 P˙ P −1 θ˜ + P −1 θ˜ = + ²φ = 0 dt m2

˜˙ d P −1 = −P −1 P˙ P −1 where the last two equalities are obtained by using θ˙ = θ, dt >˜ > ˜ ˜ ˜ ˜ = P0−1 θ(0), and, therefore, θ(t) = and ² = − θm2φ = − φm2θ . Hence, P −1 (t)θ(t) −1 ˜ −1 ˜ = P¯ P θ(0), ˜ P (t)P θ(0) and limt→∞ θ(t) which implies that limt→∞ θ(t) = θ∗ + 0

4 ˜ = ¯ P¯ P0−1 θ(0) θ.

0

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

˜ = P (t)P −1 θ(0) ˜ we have θ, θ˜ ∈ L∞ , which, together Because P (t) ≤ P0 and θ(t) 0 φ θ˜> φ with m ∈ L∞ , implies that ²m = − m and ², ²ns ∈ L∞ . Let us now consider the function ˜> −1 ˜ ˜ t) = θ P (t)θ V (θ, 2 The time derivative V˙ of V along the solution of (4.3.80) is given by θ˜> φφ> θ˜ ²2 m2 ²2 m2 2 2 = −² = − ≤0 V˙ = ²θ˜> φ + m + 2m2 2 2 which implies that V ∈ L∞ , ²m ∈ L2 ; therefore, ², ²ns ∈ L2 . From (4.3.80) we have ˙ ≤ kP k |θ|

|φ| |²m| m

T φ Because P, m , ²m ∈ L∞ and ²m ∈ L2 , we have θ˙ ∈ L∞ L2 , which completes the proof for (i), (ii), and (iii). The proof of (iv) is given in Section 4.8. 2

Remark 4.3.16 (i) We should note that the convergence rate of θ(t) to θ∗ in Theorem 4.3.4 is not guaranteed to be exponential even when φ is PE. As shown in the ˜ satisfy proof of Theorem 4.3.4 (iv) in Section 4.8, P (t), θ(t) P (t) ≤

m ¯ P0−1 m ¯ ˜ ˜ I, |θ(t)| ≤ |θ(0)|, ∀t > T0 (t − T0 )α0 (t − T0 )α0

˜ where m ¯ = supt m2 (t), i.e., |θ(t)| is guaranteed to converge to zero with 1 a speed of t . ˙ → 0 as (ii) The convergence of θ(t) to θ¯ as t → ∞ does not imply that θ(t) t → ∞ (see examples in Chapter 3). ˙ (iii) We can establish that ², θ˙ → 0 as t → ∞ if we assume that φ/m, m/m ˙ ∈ L∞ as in the case of the gradient algorithm based on the instantaneous cost. Pure Least-Squares with Covariance Resetting The so called wind-up problem of the pure least-squares algorithm is avoided by using various modifications that prevent P (t) from becoming singular.

4.3. ADAPTIVE LAWS WITH NORMALIZATION

197

One such modification is the so-called covariance resetting described by θ˙ = P ²φ P φφ> P , P˙ = − m2

P (t+ r ) = P0 = ρ0 I

(4.3.81)

where tr is the time for which λmin (P (t)) ≤ ρ1 and ρ0 > ρ1 > 0 are some design scalars. Because of (4.3.81), P (t) ≥ ρ1 I ∀t ≥ 0; therefore, P is guaranteed to be positive definite for all t ≥ 0. Strictly speaking, (4.3.81) is no longer the least-squares algorithm that we developed by setting ∇J(θ) = 0 and β = 0. It does, however, behave as a pure least-squares algorithm between resetting points. The properties of (4.3.81) are similar to those of the gradient algorithm based on the instantaneous cost. In fact, (4.3.81) may be viewed as a gradient algorithm with time-varying adaptive gain P . Theorem 4.3.5 The pure least-squares with covariance resetting algorithm (4.3.81) has the following properties: (i) ², ²ns , θ, θ˙ ∈ L∞ . (ii) ², ²ns , θ˙ ∈ L2 . (iii) If ns , φ ∈ L∞ and φ is PE then θ(t) converges exponentially to θ∗ . Proof The covariance matrix P (t) has elements that are discontinuous functions of time whose values between discontinuities are defined by the differential equation (4.3.81). At the discontinuity or resetting point tr , P (t+ r ) = P0 = ρ0 I; therefore, −1 d −1 P −1 (t+ ) = ρ I. Between discontinuities P (t) ≥ 0 , i.e., P −1 (t2 )−P −1 (t1 ) ≥ 0 r 0 dt ∀t2 ≥ t1 ≥ 0 such that tr 6∈ [t1 , t2 ], which implies that P −1 (t) ≥ ρ−1 0 I, ∀t ≥ 0. Because of the resetting, P (t) ≥ ρ1 I, ∀t ≥ 0. Therefore, (4.3.81) guarantees that −1 ρ0 I ≥ P (t) ≥ ρ1 I, ρ−1 (t) ≥ ρ−1 1 I ≥P 0 I,

∀t ≥ 0

Let us now consider the function ˜> −1 ˜ ˜ =θ P θ V (θ) 2

(4.3.82)

where P is given by (4.3.81). Because P −1 is a bounded positive definite symmetric ˜ matrix, it follows that V is decrescent and radially unbounded in the space of θ. Along the solution of (4.3.81) we have 1 d(P −1 ) ˜ ˜> −1 ˜˙ 1 d(P −1 ) ˜ V˙ = θ˜> θ + θ P θ = −²2 m2 + θ˜> θ 2 dt 2 dt

198

CHAPTER 4. ON-LINE PARAMETER ESTIMATION

Between resetting points we have from (4.3.81) that

d(P −1 ) dt

=

φφ> m2 ;

therefore,

1 (θ˜> φ)2 ²2 m2 V˙ = −²2 m2 + = − ≤0 2 m2 2

(4.3.83)

∀t ∈ [t1 , t2 ] where [t1 , t2 ] is any interval in [0, ∞) for which tr 6∈ [t1 , t2 ]. At the points of discontinuity of P , we have V (t+ r ) − V (tr ) =

1 ˜> −1 + θ (P (tr ) − P −1 (tr ))θ˜ 2

1 −1 (tr ) ≥ ρ10 I, it follows that V (t+ Because P −1 (t+ r ) − V (tr ) ≤ 0, which r ) = ρ0 I, P implies that V ≥ 0 is a nonincreasing function of time for all t ≥ 0. Hence, V ∈ L∞ and limt→∞ V (t) = V∞ < ∞. Because the points of discontinuities tr form a set of measure zero, it follows from (4.3.83) that ²m, ² ∈ L2 . From V ∈ L∞ and −1 −1 ˜ ρ−1 1 I ≥ PT (t) ≥ ρ0 I we have θ ∈ L∞ , which implies T that ², ²m ∈ L∞ . Using ²m ∈ L∞ L2 and ρ0 I ≥ P ≥ ρ1 I we have θ˙ ∈ L∞ L2 and the proof of (i) and (ii) is, therefore, complete. The proof of (iii) is very similar to the proof of Theorem 4.3.2 (iii) and is omitted. 2

Modified Least-Squares with Forgetting Factor When β > 0, the problem of P (t) becoming arbitrarily small in some directions no longer exists. In this case, however, P (t) may grow without bound >P since P˙ may satisfy P˙ > 0 because βP > 0 and the fact that P φφ is only m2 positive semidefinite. One way to avoid this complication is to modify the least-squares algorithm as follows: θ˙ = P ²φ (

P˙

=

βP − 0

P φφ> P m2

if kP (t)k ≤ R0 otherwise

(4.3.84)

where P (0) = P0 = P0> > 0, kP0 k ≤ R0 and R0 is a constant that serves as an upper bound for kP k. This modification guarantees that P ∈ L∞ and is referred to as the modified least-squares with forgetting factor. The above algorithm guarantees the same properties as the pure least-squares with covariance resetting given by Theorem 4.3.5. They can be established

4.3. ADAPTIVE LAWS WITH NORMALIZATION

199

by choosing the same Lyapunov-like function as in (4.3.82) and using the −1 identity d Pdt = −P −1 P˙ P −1 to establish dP −1 = dt

(

−βP −1 + 0

φφ> m2

if kP k ≤ R0 otherwise

where P −1 (0) = P0−1 , which leads to (

V˙ =

2 2 − ² 2m − β2 θ˜> P −1 θ˜ if kP k ≤ R0 2 2 otherwise − ² 2m

2 2 Because V˙ ≤ − ² 2m ≤ 0 and P (t) is bounded and positive definite ∀t ≥ 0, the rest of the analysis is exactly the same as in the proof of Theorem 4.3.5.

Least-Squares with Forgetting Factor and PE The covariance modifications described above are not necessary when ns , φ ∈ L∞ and φ is PE. The PE property of φ guarantees that over an interval of > time, the integral of −P φφ P is a negative definite matrix that counteracts m2 the effect of the positive definite term βP with β > 0 in the covariance equation and guarantees that P ∈ L∞ . This property is made precise by the following corollary: Corollary 4.3.2 If ns , φ ∈ L∞ and φ is PE then the recursive least-squares algorithm with forgetting factor β > 0 given by (4.3.78) and (4.3.79) guarantees that P, P −1 ∈ L∞ and that θ(t) converges exponentially to θ∗ . The proof is presented in Section 4.8. The use of the recursive least-squares algorithm with forgetting factor with φ ∈ L∞ and φ PE is appropriate in parameter estimation of stable plants where parameter convergence is the main objective. We will address such cases in Chapter 5. Let us illustrate the design of a least-squares algorithm for the same system considered in Example 4.3.1. Example 4.3.4 The system z = W (s)A sin(ωt + ϕ)

200

CHAPTER 4. ON-LINE PARAMETER ESTIMATION

where A, ϕ are to be estimated on-line is rewritten in the form z = θ∗> φ0 where θ∗ = [A1 , A2 ]> , φ0 = W (s)φ, φ = [sin ωt, cos ωt]> . The least-squares algorithm for estimating θ∗ is given by θ˙

= P ²φ0

P˙

= βP − P

φ0 φ> 0 P, m2

P (0) = ρ0 I

>

φ0 where ² = z−θ , m2 = 1 + φ> 0 φ0 and β ≥ 0, ρ0 > 0 are design constants. Because m2 φ0 is PE, no modifications are required. Let us simulate the above scheme when 2 . Figure 4.10 gives the A = 10, ϕ = 16◦ = 0.279 rad, ω = 2 rad/sec, W (s) = s+2 time response of Aˆ and ϕ, ˆ the estimate of A and ϕ, respectively, for different values of β. The simulation results indicate that the rate of convergence depends on the ˆ ϕˆ to choice of the forgetting factor β. Larger β leads to faster convergence of A, A = 10, ϕ = 0.279, respectively. 5

4.3.7

Effect of Initial Conditions

In the previous sections, we developed a wide class of on-line parameter estimators for the linear parametric model z = W (s)θ∗> ψ + η0

(4.3.85)

where η0 , the exponentially decaying to zero term that is due to initial conditions, is assumed to be equal to zero. As shown in Chapter 2 and in Section 4.3.3, η0 satisfies the equation ω˙ 0 = Λc ω0 , η0 =

C0> ω0

ω0 (0) = B0 x0 (4.3.86)

where Λc is a stable matrix, and x0 is the initial value of the plant state at t = 0.

4.3. ADAPTIVE LAWS WITH NORMALIZATION

201

2

15 Aˆ A

10

ˆϕ

1.5

ϕ

1 5

0

0.5

0

10

20 (a)

30

0 0

40

10

20 (b)

sec

30

40 sec

20 Aˆ ( t ) sin ( 2 t + ˆϕ ( t ) ) A sin ( 2 t + ϕ )

10 0 -10 -20

0

5

10

15

20 (c)

25

30

35

40

sec

Figure 4.10 Simulation results of Example 4.3.4 for the least-squares algorithm with forgetting factor and PE signals. Let us analyze the effect of η0 on the gradient algorithm θ˙ = Γ²φ z − zˆ ² = , zˆ = θ> φ m2 φ = W (s)ψ

(4.3.87)

that is developed for the model (4.3.85) with η0 = 0 in Section 4.3.5. We first express (4.3.87) in terms of the parameter error θ˜ = θ − θ∗ , i.e., ˙ θ˜ = Γ²φ

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

z − zˆ −θ˜> φ + η0 = (4.3.88) m2 m2 It is clear that η0 acts as a disturbance in the normalized estimation error and, therefore, in the adaptive law for θ. The question that arises now is whether η0 will affect the properties of (4.3.87) as described by Theorem 4.3.2. We answer this question as follows. Instead of the Lyapunov-like function ˜> −1 ˜ ˜ =θ Γ θ V (θ) 2 used in the case of η0 = 0, we propose the function ˜> −1 ˜ ˜ ω0 ) = θ Γ θ + ω > P0 ω0 V (θ, 0 2 where P0 = P0> > 0 satisfies the Lyapunov equation ² =

P0 Λc + Λ> c P0 = −γ0 I for some γ0 > 0 to be chosen. Then along the solution of (4.3.87) we have V˙ = θ˜> φ² − γ0 |ω0 |2 = −²2 m2 + ²η0 − γ0 |ω0 |2 Because η0 = C0> ω0 we have V˙

≤ −²2 m2 + |²||C0> ||ω0 | − γ0 |ω0 |2 µ

|ω0 | ²2 m2 1 − ²m − |C0> | ≤ − 2 2 m By choosing γ0 ≥

|C0> |2 2

¶2

Ã 2

− |ω0 |

|C > |2 γ0 − 0 2 2m

!

we have ²2 m2 ≤0 V˙ ≤ − 2

T φ which implies that θ˜ ∈ L∞ , ²m ∈ L2 . Because η0 ∈ L∞ L2 and m ∈ L∞ T ˙ we have ², ²m, θ ∈ L∞ L2 . Hence, (i) and (ii) of Theorem 4.3.2 also hold when η0 6= 0. In a similar manner we can show that η0 6= 0 does not affect (iii) of Theorem 4.3.2. As in every dynamic system, η0 6= 0 will affect the transient response of θ(t) depending on how fast η0 (t) → 0 as t → ∞. The above procedure can be applied to all the results of the previous sections to establish that initial conditions do not affect the established properties of the adaptive laws developed under the assumption of zero initial conditions.

4.4. ADAPTIVE LAWS WITH PROJECTION

4.4

203

Adaptive Laws with Projection

In Section 4.3 we developed a wide class of adaptive laws for estimating the constant vector θ∗ that satisfies the linear parametric model z = W (s)θ∗> ψ

(4.4.1)

by allowing θ∗ to lie anywhere in Rn . In many practical problems where θ∗ represents the parameters of a physical plant, we may have some a priori knowledge as to where θ∗ is located in Rn . This knowledge usually comes in terms of upper or lower bounds for the elements of θ∗ or in terms of a welldefined subset of Rn , etc. One would like to use such a priori information and design adaptive laws that are constrained to search for estimates of θ∗ in the set where θ∗ is located. Intuitively such a procedure may speed up convergence and reduce large transients that may occur when θ(0) is chosen to be far away from the unknown θ∗ . Another possible reason for constraining θ(t) to lie in a certain set that contains θ∗ arises in cases where θ(t) is required to have certain properties satisfied by all members of the set so that certain desired calculations that involve θ(t) are possible. Such cases do arise in adaptive control and will be discussed in later chapters. We examine how to modify the adaptive laws of Section 4.3 to handle the case of constrained parameter estimation in the following sections.

4.4.1

Gradient Algorithms with Projection

Let us start with the gradient method where the unconstrained minimization of J(θ) considered in Section 4.3.5 is extended to minimize J(θ) subject to θ ∈ S

(4.4.2)

where S is a convex set with a smooth boundary almost everywhere. Let S be given by S = {θ ∈ Rn | g(θ) ≤ 0 } (4.4.3) where g : Rn 7→ R is a smooth function. The solution of the constrained minimization problem follows from the gradient projection method discussed

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

in Appendix B and is given by    −Γ∇J

if θ ∈ S 0 or θ ∈ δ(S) and −(Γ∇J)> ∇g ≤ 0 θ˙ = Pr(−Γ∇J) = >   −Γ∇J + Γ ∇g∇g Γ∇J otherwise ∇g >Γ∇g (4.4.4) where S 0 is the interior of S, δ(S) is the boundary of S and θ(0) is chosen to be in S, i.e., θ(0) ∈ S. Let us now use (4.4.4) to modify the gradient algorithm θ˙ = −Γ∇J(θ) = Γ²φ given by (4.3.52). Because ∇J = −²φ, (4.4.4) becomes 4

   Γ²φ

if θ ∈ S 0 or if θ ∈ δ(S) and (Γ²φ)> ∇g ≤ 0 θ˙ = Pr(Γ²φ) = >   Γ²φ − Γ ∇g∇g Γ²φ otherwise ∇g > Γ∇g (4.4.5) where θ(0) ∈ S. In a similar manner we can modify the integral adaptive law (4.3.60) by substituting ∇J = R(t)θ + Q(t) in (4.4.4). The principal question we need to ask ourselves at this stage is whether projection will destroy the properties of the unconstrained adaptive laws developed in Section 4.3.5. This question is answered by the following Theorem. Theorem 4.4.1 The gradient adaptive laws of Section 4.3.5 with the projection modification given by (4.4.4) retain all their properties that are established in the absence of projection and in addition guarantee that θ ∈ S ∀t ≥ 0 provided θ(0) = θ0 ∈ S and θ∗ ∈ S. Proof It follows from (4.4.4) that whenever θ ∈ δ(S) we have θ˙> ∇g ≤ 0, which implies that the vector θ˙ points either inside S or along the tangent plane of δ(S) at point θ. Because θ(0) = θ0 ∈ S, it follows that θ(t) will never leave S, i.e., θ(t) ∈ S ∀t ≥ 0. The adaptive law (4.4.4) has the same form as the one without projection except for the additional term ( ∇g∇g > Γ ∇g if θ ∈ δ(S) and −(Γ∇J)> ∇g > 0 > Γ∇g Γ∇J Q= 0 otherwise ˙ If we use the same function V as in the unconstrained case in the expression for θ. to analyze the adaptive law with projection, the time derivative V˙ of V will have

4.4. ADAPTIVE LAWS WITH PROJECTION

205

the additional term ( θ˜> Γ−1 Q =

∇g∇g > θ˜> ∇g > Γ∇g Γ∇J 0

if θ ∈ δ(S) and −(Γ∇J)> ∇g > 0 otherwise

Because of the convex property of S and the assumption that θ∗ ∈ S, we have θ˜> ∇g = (θ − θ∗ )> ∇g ≥ 0 when θ ∈ δ(S). Because ∇g > Γ∇J = (Γ∇J)> ∇g < 0 for θ ∈ δ(S) and −(Γ∇J)> ∇g > 0, it follows that θ˜> Γ−1 Q ≤ 0. Therefore, the term θ˜> Γ−1 Q introduced by the projection can only make V˙ more negative and does not affect the results developed from the properties of V, V˙ . Furthermore, the L2 properties of θ˙ will not be affected by projection because, with or without projection, θ˙ can be shown to satisfy ˙ 2 ≤ c|Γ∇J|2 |θ| for some constant c ∈ R+ .

2

The projection modification (4.4.4) holds also for the adaptive laws based on the SPR-Lyapunov design approach even though these adaptive laws are not derived from the constrained optimization problem defined in (4.4.2). The reason is that the adaptive law (4.3.35) based on the SPR-Lyapunov approach and the gradient algorithm based on the instantaneous cost have the same form, i.e., θ˙ = Γ²φ where ², φ are, of course, not the same signals in general. Therefore, by substituting for −Γ∇J = Γ²φ in (4.4.4), we can obtain the SPR-Lyapunov based adaptive law with projection. We can establish, as done in Theorem 4.4.1, that the adaptive laws based on the SPR-Lyapunov design approach with the projection modification retain their original properties established in the absence of projection. Remark 4.4.1 We should emphasize that the set S is required to be convex. The convexity of S helps in establishing the fact that the projection does not alter the properties of the adaptive laws established without projection. Projection, however, may affect the transient behavior of the adaptive law. Let us now consider some examples of constrained parameter estimation.

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

Example 4.4.1 Consider the linear parametric model z = θ∗> φ where θ∗ = [θ1∗ , θ2∗ , . . . , θn∗ ]> is an unknown constant vector, θ1∗ is known to satisfy |θ1∗ | ≥ ρ0 > 0 for some known constant ρ0 , sgn(θ1∗ ) is known, and z, φ can be measured. We would like to use this a priori information about θ1∗ and constrain its estimation to always be inside a convex set S which contains θ∗ and is defined as 4 S = {θ ∈ Rn | g(θ) = ρ0 − θ1 sgn(θ1∗ ) ≤ 0 } The gradient algorithm with    Γ²φ ˙θ = >   Γ²φ − Γ ∇g∇g Γ²φ ∇g > Γ∇g

projection becomes if ρ0 − θ1 sgn(θ1∗ ) < 0 or if ρ0 − θ1 sgn(θ1∗ ) = 0 and (Γ²φ)> ∇g ≤ 0 otherwise

(4.4.6)

>

φ where ² = z−θ and θ1 (0) satisfies ρ0 − θ1 (0)sgn(θ1∗ ) < 0. For simplicity, let us m2 choose Γ = diag{γ1 , Γ0 } where γ1 > 0 is a scalar and Γ0 = Γ> 0 > 0, and partition 1 > > > φ, θ as φ = [φ1 , φ> 0 ] , θ = [θ1 , θ0 ] where φ1 , θ1 ∈ R . Because

∇g = [−sgn(θ1∗ ), 0, . . . , 0]> it follows from (4.4.6) that   γ1 ²φ1 if θ1 sgn(θ1∗ ) > ρ0 or if θ1 sgn(θ1∗ ) = ρ0 and γ1 φ1 ²sgn(θ1∗ ) ≥ 0 θ˙1 =  0 otherwise θ˙0 = Γ0 ²φ0 where θ0 (0) is arbitrary and θ1 (0) satisfies θ1 (0)sgn(θ1∗ ) > ρ0 .

4.4.2

(4.4.7)

5

Least-Squares with Projection

The gradient projection method can also be adopted in the case of the least squares algorithm θ˙ = P ²φ P˙

= βP − P

φφ> P, m2

P (0) = P0 = Q−1 0

(4.4.8)

developed in Section 4.3.6 by viewing (4.4.8) as a gradient algorithm with time varying scaling matrix P and ²φ as the gradient of some cost function

4.4. ADAPTIVE LAWS WITH PROJECTION

207

J. If S = {θ ∈ Rn | g(θ) ≤ 0 } is the convex set for constrained estimation, then (4.4.8) is modified as    P ²φ

if θ ∈ S 0 or if θ ∈ δ(S) and (P ²φ)>∇g ≤ 0 θ˙ = Pr(P ²φ) = >   P ²φ − P ∇g∇g P ²φ otherwise ∇g > P ∇g (4.4.9) where θ(0) ∈ S and P˙ =

 φφ>   βP − P m2 P  

0

if θ ∈ S 0 or if θ ∈ δ(S) and (P ²φ)> ∇g ≤ 0 otherwise

(4.4.10)

where P (0) = P0 = P0> > 0. It can be shown as in Section 4.4.1 that the least-squares with projection has the same properties as the corresponding least-squares without projection. The equation for the covariance matrix P is modified so that at the point of projection on the boundary of S, P is a constant matrix, and, therefore, the adaptive law at that point is a gradient algorithm with constant scaling that justifies the use of the gradient projection method explained in Appendix B. Example 4.4.2 Let us now consider a case that often arises in adaptive control in the context of robustness. We would like to constrain the estimates θ(t) of θ∗ to remain inside a bounded convex set S. Let us choose S as ¯ © ª S = θ ∈ Rn ¯θ> θ − M02 ≤ 0 for some known constant M0 such that |θ∗ | ≤ M0 . The set S represents a sphere in Rn centered at θ = 0 and of radius M0 . We have ∇g = 2θ and the least-squares algorithm    P ²φ ˙θ = ´ ³   I − P>θθ> P ²φ θ Pθ

with projection becomes if θ> θ < M02 or if θ> θ = M02 and (P ²φ)> θ ≤ 0 otherwise

(4.4.11)

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

where θ(0) satisfies θ> (0)θ(0) ≤ M02 and P is given by P˙ =

 >  βP − P φφ m2 P 

0

if θ> θ < M02 or if θ> θ = M02 and (P ²φ)> θ ≤ 0 otherwise

(4.4.12)

Because P = P > > 0 and θ> P θ > 0 when θ> θ = M02 , no division by zero occurs in (4.4.11). 5

4.5

Bilinear Parametric Model

As shown in Chapter 2, a certain class of plants can be parameterized in terms of their desired controller parameters that are related to the plant parameters via a Diophantine equation. Such parameterizations and their related estimation problem arise in direct adaptive control, and in particular, direct MRAC, which is discussed in Chapter 6. In these cases, θ∗ , as shown in Chapter 2, appears in the form z = W (s)[ρ∗ (θ∗> ψ + z0 )]

(4.5.1)

where ρ∗ is an unknown constant; z, ψ, z0 are signals that can be measured and W (s) is a known proper transfer function with stable poles. Because the unknown parameters ρ∗ , θ∗ appear in a special bilinear form, we refer to (4.5.1) as the bilinear parametric model. The procedure of Section 4.3 for estimating θ∗ in a linear model extends to (4.5.1) with minor modifications when the sgn(ρ∗ ) is known or when sgn(ρ∗ ) and a lower bound ρ0 of |ρ∗ | are known. When the sgn(ρ∗ ) is unknown the design and analysis of the adaptive laws require some additional modifications and stability arguments. We treat each case of known and unknown sgn(ρ∗ ), ρ0 separately.

4.5.1

Known Sign of ρ∗

The SPR-Lyapunov design approach and the gradient method with an instantaneous cost function discussed in the linear parametric case extend to the bilinear one in a rather straightforward manner.

4.5. BILINEAR PARAMETRIC MODEL

209

Let us start with the SPR-Lyapunov design approach. We rewrite (4.5.1) in the form z = W (s)L(s)ρ∗ (θ∗> φ + z1 ) (4.5.2) where z1 = L−1 (s)z0 , φ = L−1 (s)ψ and L(s) is chosen so that L−1 (s) is proper and stable and W L is proper and SPR. The estimate zˆ of z and the normalized estimation error are generated as zˆ = W (s)L(s)ρ(θ> φ + z1 )

(4.5.3)

² = z − zˆ − W (s)L(s)² n2s

(4.5.4)

where ns is designed to satisfy φ z1 , ∈ L∞ , m2 = 1 + n2s m m

(A2)

and ρ(t), θ(t) are the estimates of ρ∗ , θ∗ at time t, respectively. Letting 4 4 ρ˜ = ρ − ρ∗ , θ˜ = θ − θ∗ , it follows from (4.5.2) to (4.5.4) that ² = W (s)L(s)[ρ∗ θ∗> φ − ρ˜z1 − ρθ> φ − ²n2s ] Now ρ∗ θ∗> φ − ρθ> φ = ρ∗ θ∗> φ − ρ∗ θ> φ + ρ∗ θ> φ − ρθ> φ = −ρ∗ θ˜> φ − ρ˜θ> φ and, therefore, ² = W (s)L(s)[−ρ∗ θ˜> φ − ρ˜ξ − ²n2s ], ξ = θ> φ + z1

(4.5.5)

A minimal state representation of (4.5.5) is given by e˙ = Ac e + Bc (−ρ∗ θ˜> φ − ρ˜ξ − ²n2s ) ² = Cc> e

(4.5.6)

where Cc> (sI − Ac )−1 Bc = W (s)L(s) is SPR. The adaptive law is now developed by considering the Lyapunov-like function > 2 ˜> −1 ˜ ˜ ρ˜) = e Pc e + |ρ∗ | θ Γ θ + ρ˜ V (θ, 2 2 2γ

where Pc = Pc> > 0 satisfies the algebraic equations given by (4.3.32) that are implied by the KYL Lemma, and Γ = Γ> > 0, γ > 0. Along the solution of (4.5.6), we have e> qq > e ν > ˙ ρ˜ρ˜˙ V˙ = − − e Lc e − ρ∗ ²θ˜> φ − ²˜ ρξ − ²2 n2s + |ρ∗ |θ˜> Γ−1 θ˜ + 2 2 γ

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

∗ ∗ ∗ where ν > 0, Lc = L> c > 0. Because ρ = |ρ |sgn(ρ ) it follows that by choosing ˙ θ˜ = θ˙ = Γ²φsgn(ρ∗ ) (4.5.7) ρ˜˙ = ρ˙ = γ²ξ

we have

e> qq > e ν > V˙ = − − e Lc e − ²2 n2s ≤ 0 2 2 The rest of the analysis continues as in the case of the linear model. We summarize the properties of the bilinear adaptive law by the following theorem.

Theorem 4.5.1 The adaptive law (4.5.7) guarantees that (i) (ii) (iii) (iv)

², θ, ρ ∈ L∞ . ˙ ρ˙ ∈ L2 . ², ²ns , θ, If φ, φ˙ ∈ L∞ , φ is PE and ξ ∈ L2 , then θ(t) converges to θ∗ as t → ∞. If ξ ∈ L2 , the estimate ρ converges to a constant ρ¯ independent of the properties of φ.

Proof The proof of (i) and (ii) follows directly from the properties of V, V˙ by following the same procedure as in the linear parametric model case and is left as an exercise for the reader. The proof of (iii) is established by using the results of Corollary 4.3.1 to show that the homogeneous part of (4.5.6) with ρ˜ξ treated as an external input together with the equation of θ˜ in (4.5.7) form an e.s. system. Because ρ˜ξ ∈ L2 and Ac is stable, it follows that e, θ˜ → 0 as t → ∞. The details of the proof are given in Section 4.8. The proof of (iv) follows from ², ξ ∈ L2 and the inequality Z

Z

t

|ρ|dτ ˙ ≤γ 0

µZ

t

|²ξ|dτ ≤ γ 0

∞

2

¶ 12 µZ

∞

² dτ 0

2

ξ dτ

¶ 21 <∞

0

which implies that ρ˙ ∈ L1 . Therefore, we conclude that ρ(t) has a limit ρ¯, i.e., limt→∞ ρ(t) = ρ¯. 2

The lack of convergence of ρ to ρ∗ is due to ξ ∈ L2 . If, however, φ, ξ 4 are such that φα = [φ> , ξ]> is PE, then we can establish by following the ˜ ρ˜ converge to zero same approach as in the proof of Corollary 4.3.1 that θ,

4.5. BILINEAR PARAMETRIC MODEL

211

exponentially fast. For ξ ∈ L2 , the vector φα cannot be PE even when φ is PE. For the gradient method we rewrite (4.5.1) as z = ρ∗ (θ∗> φ + z1 )

(4.5.8)

where z1 = W (s)z0 , φ = W (s)ψ. Then the estimate zˆ of z and the normalized estimation error ² are given by zˆ = ρ(θ> φ + z1 ) z − ρ(θ> φ + z1 ) z − zˆ ² = = m2 m2

(4.5.9)

where n2s is chosen so that φ z1 , ∈ L∞ , m2 = 1 + n2s m m As in the case of the linear model, we consider the cost function J(ρ, θ) =

(A2)

² 2 m2 (z − ρ∗ θ> φ − ρξ + ρ∗ ξ − ρ∗ z1 )2 = 2 2m2

where ξ = θ> φ + z1 and the second equality is obtained by using the identity −ρ(θ> φ + z1 ) = −ρξ − ρ∗ θ> φ + ρ∗ ξ − ρ∗ z1 . Strictly speaking J(ρ, θ) is not a convex function of ρ, θ over Rn+1 because of the dependence of ξ on θ. Let us, however, ignore this dependence and treat ξ as an independent function of time. Using the gradient method and treating ξ as an arbitrary function of time, we obtain θ˙ = Γ1 ρ∗ ²φ, ρ˙ = γ²ξ (4.5.10) where Γ1 = Γ> 1 > 0, γ > 0 are the adaptive gains. The adaptive law (4.5.10) cannot be implemented due to the unknown ρ∗ . We go around this difficulty as follows: Because Γ1 is arbitrary, we assume that Γ1 = |ρΓ∗ | for some other arbitrary matrix Γ = Γ> > 0 and use it together with ρ∗ = |ρ∗ |sgn(ρ∗ ) to get rid of the unknown parameter ρ∗ , i.e., Γ1 ρ∗ = |ρΓ∗ | ρ∗ = Γsgn(ρ∗ ) leading to θ˙ = Γ²φsgn(ρ∗ ), ρ˙ = γ²ξ (4.5.11) which is implementable. The properties of (4.5.11) are given by the following theorem.

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

Theorem 4.5.2 The adaptive law (4.5.11) guarantees that ˙ ρ, ρ˙ ∈ L∞ . (i) ², ²ns , θ, θ, ˙ ρ˙ ∈ L2 . (ii) ², ²ns , θ, (iii) If ns , φ ∈ L∞ , φ is PE and ξ ∈ L2 , then θ(t) converges to θ∗ as t → ∞. (iv) If ξ ∈ L2 , then ρ converges to a constant ρ¯ as t → ∞ independent of the properties of φ. The proof follows from that of the linear parametric model and of Theorem 4.5.1, and is left as an exercise for the reader. The extension of the integral adaptive law and least-squares algorithm to the bilinear parametric model is more complicated and difficult to implement due to the appearance of the unknown ρ∗ in the adaptive laws. This problem is avoided by assuming the knowledge of a lower bound for |ρ∗ | in addition to sgn(ρ∗ ) as discussed in the next section.

4.5.2

Sign of ρ∗ and Lower Bound ρ0 Are Known

The complications with the bilinearity in (4.5.1) are avoided if we rewrite (4.5.1) in the form of the linear parametric model z = θ¯∗> φ¯

(4.5.12)

where θ¯∗ = [θ¯1∗ , θ¯2∗> ]> , φ¯ = [z1 , φ> ]> , and θ¯1∗ = ρ∗ , θ¯2∗ = ρ∗ θ∗ . We can now ¯ of θ¯∗ at each use the methods of Section 4.3 to generate the estimate θ(t) time t. From the estimate θ¯ = [θ¯1 , θ¯2> ]> of θ¯∗ , we calculate the estimate ρ, θ of ρ∗ , θ∗ as follows: θ¯2 (t) (4.5.13) ρ(t) = θ¯1 (t), θ(t) = ¯ θ1 (t) The possibility of division by zero or a small number in (4.5.13) is avoided by constraining the estimate of θ¯1 to satisfy |θ¯1 (t)| ≥ ρ0 > 0 for some ρ0 ≤ |ρ∗ |. This is achieved by using the gradient projection method and assuming that ρ0 and sgn(ρ∗ ) are known. We illustrate the design of such a gradient algorithm as follows: By considering (4.5.12) and following the procedure of Section 4.3, we generate ¯ ² = z − zˆ (4.5.14) zˆ = θ¯> φ, m2

4.5. BILINEAR PARAMETRIC MODEL

213

¯ ¯ The where m2 = 1 + n2s and ns is chosen so that φ/m ∈ L∞ , e.g. n2s = φ¯> φ. adaptive law is developed by using the gradient projection method to solve the constrained minimization problem ¯> ¯ 2 ¯ = min (z − θ φ) min J(θ) 2m2 θ¯ θ¯ subject to ρ0 − θ¯1 sgn(ρ∗ ) ≤ 0 i.e.,  ¯   Γ²φ

if ρ0 − θ¯1 sgn(ρ∗ ) < 0 ¯ > ∇g ≤ 0 or if ρ0 − θ¯1 sgn(ρ∗ ) = 0 and (Γ²φ) θ¯˙ = >   Γ²φ ¯ − Γ ∇g∇g Γ²φ¯ otherwise ∇g > Γ∇g (4.5.15) ¯ = ρ0 − θ¯1 sgn(ρ∗ ). For simplicity, let us assume that Γ = where g(θ) diag{γ1 , Γ2 } where γ1 > 0 is a scalar and Γ2 = Γ> 2 > 0 and simplify the expressions in (4.5.15). Because ∇g = [−sgn(ρ∗ ), 0, . . . , 0]> it follows from (4.5.15) that θ¯˙ 1 =

 ¯1   γ1 ²φ   0

if θ¯1 sgn(ρ∗ ) > ρ0 or if θ¯1 sgn(ρ∗ ) = ρ0 and −γ1 φ¯1 ²sgn(ρ∗ ) ≤ 0 otherwise

(4.5.16)

where θ¯1 (0) satisfies θ¯1 (0)sgn(ρ∗ ) ≥ ρ0 , and θ¯˙ 2 = Γ2 ²φ¯2

(4.5.17)

4 4 where φ¯1 = z1 , φ¯2 = φ. Because θ¯1 (t) is guaranteed by the projection to satisfy |θ¯1 (t)| ≥ ρ0 > 0, the estimate ρ(t), θ(t) can be calculated using (4.5.13) without the possibility of division by zero. The properties of the adaptive law (4.5.16), (4.5.17) with (4.5.13) are summarized by the following theorem.

Theorem 4.5.3 The adaptive law described by (4.5.13), (4.5.16), (4.5.17) guarantees that

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

(i) ², ²ns , ρ, θ, ρ, ˙ θ˙ ∈ L∞ . ˙ (ii) ², ²ns , ρ, ˙ θ ∈ L2 . ¯ θ, ρ converge to θ¯∗ , θ∗ , ρ∗ , respec(iii) If ns , φ¯ ∈ L∞ and φ¯ is PE, then θ, tively, exponentially fast. Proof Consider the Lyapunov-like function V = 4 where ˜θ¯1 = θ¯1 − θ¯1∗ , ˜θ¯2 have  2 2   −² m V˙ =   ˜θ¯> φ¯ ² 2 2

˜θ¯> Γ−1 ˜θ¯ ˜θ¯2 2 1 + 2 2 2γ1 2

4

= θ¯2 − θ¯2∗ . Then along the solution of (4.5.16), (4.5.17), we if θ¯1 sgn(ρ∗ ) > ρ0 or if θ¯1 sgn(ρ∗ ) = ρ0 and −γ1 φ¯1 ²sgn(ρ∗ ) ≤ 0 if θ¯1 sgn(ρ∗ ) = ρ0 and −γ1 φ¯1 ²sgn(ρ∗ ) > 0

(4.5.18)

> > > Because ²m2 = −˜θ¯ φ¯ = −˜θ¯1 φ¯1 − ˜θ¯2 φ¯2 , we have ˜θ¯2 φ¯2 ² = −²2 m2 − ˜θ¯1 ²φ¯1 . For θ¯1 sgn(ρ∗ ) = ρ0 (i.e., θ¯1 = ρ0 sgn(ρ∗ )) and −γ1 φ¯1 ²sgn(ρ∗ ) > 0, we have ˜θ¯1 φ¯1 ² = (ρ0 sgn(ρ∗ ) − |ρ∗ |sgn(ρ∗ ))φ¯1 ² = (ρ0 − |ρ∗ |)sgn(ρ∗ )φ¯1 ² > 0 (because ρ0 − |ρ∗ | < 0 and sgn(ρ∗ )φ¯1 ² < 0), which implies that >

²˜θ¯2 φ¯2 = −²2 m2 − ²φ¯1 (ρ0 − |ρ∗ |)sgn(ρ∗ ) < −²2 m2 Therefore, projection introduces the additional term −²φ¯1 (ρ0 − |ρ∗ |)sgn(ρ∗ ) that > can only make V˙ more negative. Substituting for ²˜θ¯2 φ¯2 < −²2 m2 in (4.5.18) we have V˙ ≤ −²2 m2 T which implies that θ¯1 , θ¯2 ∈ L∞ ; ², ²ns ∈TL∞ L2 . ¯ Because φ/m ∈ L∞ and ²m ∈ L∞ L2 , it follows from (4.5.16), (4.5.17) that T ˙θ ∈ L ¯ L2 , i = 1, 2. i ∞ ˙ ¯ ˙ ¯ ¯ Using (4.5.13), we have ρ˙ = θ¯˙ 1 , θ˙ = θθ¯12 − θθ1¯2θ2 , which, together with θ¯˙ i ∈ 1 T T L∞ L2 , i = 1, 2, |θ¯1 | ≥ ρ0 > 0, imply that ρ, ˙ θ˙ ∈ L∞ L2 . The convergence of θ¯ to θ¯∗ follows from that of Theorem 4.3.2 (iii) and the fact that projection can only make V˙ more negative. The convergence of θ, ρ to θ∗ , ρ∗ follows from that of θ¯∗ , equation (4.5.13), assumption |θ¯1∗ | ≥ ρ0 and |θ¯1 (t)| > ρ0 ∀t ≥ 0. 2

In a similar manner one may use (4.5.12) and (4.5.13) to derive adaptive laws using the integral cost function and least-squares with θ¯1 constrained to satisfy |θ¯1 (t)| ≥ ρ0 > 0 ∀t ≥ 0.

4.5. BILINEAR PARAMETRIC MODEL

4.5.3

215

Unknown Sign of ρ∗

The problem of designing adaptive laws for the bilinear model (4.5.1) with sgn(ρ∗ ) unknown was motivated by Morse [155] in the context of MRAC where such an estimation problem arises. Morse conjectured that the sgn(ρ∗ ) is necessary for designing appropriate adaptive control laws used to solve the stabilization problem in MRAC. Nussbaum [179] used a simple example to show that although Morse’s conjecture was valid for a class of adaptive control laws, the sgn(ρ∗ ) is not necessary for stabilization in MRAC if a different class of adaptive or control laws is employed. This led to a series of results on MRAC [137, 156, 157, 167, 236] where the sgn(ρ∗ ) was no longer required to be known. In our case, we use the techniques of [179] to develop adaptive laws for the bilinear model (4.5.1) that do not require the knowledge of sgn(ρ∗ ). The design and analysis of these adaptive laws is motivated purely from stability arguments that differ from those used when sgn(ρ∗ ) is known. We start with the parametric model z = ρ∗ (θ∗> φ + z1 ) and generate zˆ, ² as zˆ = N (x)ρ(θ> φ + z1 ), ² = where m2 = 1 + n2s is designed so that

φ z1 m, m

z − zˆ m2

∈ L∞ ,

N (x) = x2 cos x

(4.5.19)

x is generated from x=w+

ρ2 ; 2γ

w˙ = ²2 m2 ,

w(0) = 0

(4.5.20)

where γ > 0. The following adaptive laws are proposed for generating ρ, θ: θ˙ = N (x)Γ²φ,

ρ˙ = N (x)γ²ξ

(4.5.21)

where ξ = θ> φ + z1 . The function N (x) plays the role of an adaptive gain in (4.5.21) and has often been referred to as the Nussbaum gain. Roughly

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

speaking, N (x) accounts for the unknown sign of ρ∗ by changing the sign of the vector field of θ, ρ periodically with respect to the signal x. The design of (4.5.19) to (4.5.21) is motivated from the analysis that is given in the proof of the following theorem, which states the properties of (4.5.19) to (4.5.21). Theorem 4.5.4 The adaptive law (4.5.19) to (4.5.21) guarantees that (i) (ii)

x, w, θ, ρ ∈ L∞ ˙ ρ˙ ∈ L∞ T L2 ², ²ns , θ,

Proof We start by expressing the normalized estimation error ² in terms of the ˜ i.e., parameter error θ, ²m2

= z − zˆ = ρ∗ θ∗> φ + ρ∗ z1 − ρ∗ θ> φ − ρ∗ z1 + ρ∗ ξ − N (x)ρξ = −ρ∗ θ˜> φ + ρ∗ ξ − N (x)ρξ

(4.5.22)

We choose the function

θ˜> Γ−1 θ˜ 2 whose time derivative along the solution of (4.5.21) is given by V =

V˙ = θ˜> φ²N (x) Substituting θ˜> φ =

1 2 ∗ ρ∗ [−²m + ρ ξ − N (x)ρξ]

(4.5.23)

from (4.5.22) into (4.5.23), we obtain

N (x) V˙ = − ∗ [²2 m2 − ρ∗ ²ξ + N (x)ρ²ξ] ρ or

· ¸ N (x) 2 2 ρρ˙ ρ˙ ˙ V =− ∗ ² m + + ρ γ γ

From (4.5.20), we have x˙ = ²2 m2 +

ρρ˙ γ

which we use to rewrite V˙ as

N (x)x˙ ρ˙ V˙ = − + ρ∗ γ

(4.5.24)

Integrating (4.5.24) on both sides we obtain ρ(t) − ρ(0) 1 V (t) − V (0) = − ∗ γ ρ

Z

x(t)

N (σ)dσ 0

4.6. HYBRID ADAPTIVE LAWS

217

R x(t) Because N (x) = x2 cos x and 0 σ 2 cos σdσ = 2x cos x + (x2 − 2) sin x, it follows that ρ(t) − ρ(0) 1 V (t) − V (0) = − ∗ [2x cos x + (x2 − 2) sin x] (4.5.25) γ ρ 2

ρ and w ≥ 0, we conclude that x(t) ≥ 0. Examination of (4.5.25) From x = w + 2γ shows that for large x, the term x2 sin x dominates the right-hand side of (4.5.25) and oscillates between −x2 and x2 . Because

V (t) = V (0) +

ρ(t) − ρ(0) 1 x2 sin x − ∗ [2x cos x − 2 sin x] − γ ρ ρ∗

and V (t) ≥ 0, it follows that x has to be bounded, otherwise the inequality V ≥ 0 will be violated for large x. Bounded x implies that V, w, ρ, θ ∈ L∞ . Because w(t) = Rt 2 2 ² m dt is a nondecreasing function bounded from above, the limt→∞ w(t) = R0∞ 2 2 ² m dt exists and is finite which implies that ²m ∈ L2 , i.e., ², ²ns ∈ L2 . The 0 rest of the proof follows directly as in the case of the linear parametric model and is omitted. 2

4.6

Hybrid Adaptive Laws

The adaptive laws developed in Sections 4.3 to 4.5 update the estimate θ(t) of the unknown parameter vector θ∗ continuously with time, i.e., at each time t we have a new estimate. For computational and robustness reasons it may be desirable to update the estimates only at specific instants of time tk where {tk } is an unbounded monotonically increasing sequence in R+ . Let tk = kTs where Ts = tk+1 − tk is the “sampling” period and k = 0, 1, 2, . . . ,. Consider the design of an adaptive law that generates the estimate of the unknown θ∗ at the discrete instances of time t = 0, Ts , 2Ts , · · · . We can develop such an adaptive law for the gradient algorithm θ˙ = Γ²φ z − zˆ m2 where z is the output of the linear parametric model ²=

z = θ∗T φ

(4.6.1) (4.6.2)

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

and zˆ = θ> φ

(4.6.3)

Integrating (4.6.1) from tk = kTs to tk+1 = (k + 1)Ts we have θk+1 = θk + Γ

Z tk+1 tk

²(τ )φ(τ )dτ,

θ0 = θ(0), k = 0, 1, 2, . . .

(4.6.4)

4

where θk = θ(tk ). Equation (4.6.4) generates a sequence of estimates, i.e., θ0 = θ(0), θ1 = θ(Ts ), θ2 = θ(2Ts ), . . . , θk = θ(kTs ) of θ∗ . If we now use (4.6.4) instead of (4.6.1) to estimate θ∗ , the error ² and zˆ have to be generated using θk instead of θ(t), i.e., zˆ(t) = θk> φ(t), ²(t) =

z(t) − zˆ(t) , m2 (t)

∀t ∈ [tk , tk+1 ]

(4.6.5)

We refer to (4.6.4), (4.6.5) as the hybrid adaptive law. The following theorem establishes its stability properties. Theorem 4.6.1 Let m, Ts , Γ be chosen so that (a) (b)

φ> φ m2

≤ 1, m ≥ 1 2 − Ts λm ≥ γ for some γ > 0

where λm = λmax (Γ). Then the hybrid adaptive law (4.6.4), (4.6.5) guarantees that (i) θk ∈ l∞ . T (ii) ∆θk ∈ l2 ; ², ²m ∈ L∞ L2 , where ∆θk = θk+1 − θk . (iii) If m, φ ∈ L∞ and φ is PE, then θk → θ∗ as k → ∞ exponentially fast. Proof As in the stability proof for the continuous adaptive laws, we evaluate the rate of change of the Lyapunov-like function V (k) = θ˜k> Γ−1 θ˜k

(4.6.6)

4 along the trajectory generated by (4.6.4) and (4.6.5), where θ˜k = θk − θ∗ . Notice that ∆V (k) = (2θ˜k + ∆θk )> Γ−1 ∆θk

4.6. HYBRID ADAPTIVE LAWS

219

where ∆V (k) = V (k + 1) − V (k). Using (4.6.4) we obtain Z ∆V (k) = 2θ˜k>

µZ

tk+1

¶>

tk+1

²(τ )φ(τ )dτ +

²(τ )φ(τ )dτ

tk

Z

tk+1

Γ

²(τ )φ(τ )dτ

tk

tk

Because ²m2 = −θ˜k> φ(t) and λm ≥ kΓk, we have Z

tk+1

∆V (k) ≤ −2

µZ ²2 (τ )m2 (τ )dτ + λm

tk+1

|φ(τ )| dτ m(τ )

|²(τ )m(τ )|

tk

tk

¶2 (4.6.7)

Using the Schwartz inequality, we can establish that µZ

tk+1

|²(τ )m(τ )| tk

|φ(τ )| dτ m(τ )

¶2

Z

tk+1

≤

Z ²2 (τ )m2 (τ )dτ

tk

tk

Z

≤

tk+1

tk+1

Ts

µ

|φ(τ )| m(τ )

²2 (τ )m2 (τ )dτ

¶2 dτ (4.6.8)

tk

where the last inequality follows from assumption (a). Using (4.6.8) in (4.6.7), it follows that Z tk+1 ∆V (k) ≤ −(2 − Ts λm ) ²2 (τ )m2 (τ )dτ (4.6.9) tk

Therefore, if 2 − Ts λm > γ for some γ > 0, we have ∆V (k) ≤ 0, which implies that V (k) is a nonincreasing function and thus the boundedness of V (k), θ˜k and θk follows. From (4.6.9), one can easily see that Z tk+1 V (0) − V (k + 1) (4.6.10) ²2 (τ )m2 (τ )dτ ≤ (2 − Ts λm ) 0 We T can establish that limk→∞ V (k + 1) exists and, from (4.6.10), that ²m ∈ T L∞ L2 . Because m ≥ 1, it follows immediately that ² ∈ L∞ L2 . Similarly, we can obtain that Z tk+1 > 2 ∆θk ∆θk ≤ Ts λm ²2 (τ )m2 (τ )dτ (4.6.11) tk

by using the Schwartz inequality and condition (a) of the theorem. Therefore, ∞ X k=1

Z ∆θk> ∆θk ≤ Ts λ2m

∞

²2 (τ )m2 (τ )dτ < ∞

0

which implies ∆θk ∈ l2 and, thus, completes the proof for (i) and (ii). The proof for (iii) is relatively involved and is given in Section 4.8.

2

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

For additional reading on hybrid adaptation the reader is referred to [60] where the term hybrid was first introduced and to [172, 173] where different hybrid algorithms are introduced and analyzed. The hybrid adaptive law (4.6.4) may be modified to guarantee that θk ∈ Rn belongs to a certain convex subset S of Rn by using the gradient projection method. That is, if S is defined as n

S = θ ∈ Rn | θ> θ ≤ M02

o

then the hybrid adaptive law (4.6.4) with projection becomes θ¯k+1 = θk + Γ θk+1 =

Z tk+1

( ¯ θk+1

tk

θ¯k+1 M |θ¯k+1 | 0

²(τ )φ(τ )dτ if if

θ¯k+1 ∈ S θ¯k+1 6∈ S

and θ0 ∈ S. As in the continuous-time case, it can be shown that the hybrid adaptive law with projection has the same properties as those of (4.6.4). In addition it guarantees that θk ∈ S, ∀k ≥ 0. The details of this analysis are left as an exercise for the reader.

4.7

Summary of Adaptive Laws

In this section, we present tables with the adaptive laws developed in the previous sections together with their properties.

4.8

Parameter Convergence Proofs

In this section, we present the proofs of the theorems and corollaries of the previous sections that deal with parameter convergence. These proofs are useful for the reader who is interested in studying the behavior and convergence properties of the parameter estimates. They can be omitted by the reader whose interest is mainly on adaptive control where parameter convergence is not part of the control objective.

4.8.1

Useful Lemmas

The following two lemmas are used in the proofs of corollaries and theorems presented in this sections.

4.8. PARAMETER CONVERGENCE PROOFS

221

Table 4.1 Adaptive law based on SPR-Lyapunov design approach Parametric model

z = W (s)θ∗> ψ

Parametric model rewritten

z = W (s)L(s)θ∗> φ, φ = L−1 (s)ψ

Estimation model

zˆ = W (s)L(s)θ> φ

Normalized estimation error

² = z − zˆ − W (s)L(s)²n2s

Adaptive law

θ˙ = Γ²φ

Design variables

L−1 (s) proper and stable; W (s)L(s) proper and φ SPR; m2 = 1 + n2s and ns chosen so that m ∈ L∞ (e. g., n2s = αφ> φ for some α > 0)

Properties

(i) ², θ ∈ L∞ ; (ii) ², ²ns , θ˙ ∈ L2

Lemma 4.8.1 (Uniform Complete Observability (UCO) with Output Injection). Assume that there exists constants ν > 0, kν ≥ 0 such that for all t0 ≥ 0, K(t) ∈ Rn×l satisfies the inequality Z t0 +ν |K(τ )|2 dτ ≤ kν (4.8.1) t0

∀t ≥ 0 and some constants k0 , ν > 0. Then (C, A), where C ∈ Rn×l , A ∈ Rn×n , is a UCO pair if and only if (C, A + KC > ) is a UCO pair. Proof We show that if there exist positive constants β1 , β2 > 0 such that the observability grammian N (t0 , t0 + ν) of the system (C, A) satisfies β1 I ≤ N (t0 , t0 + ν) ≤ β2 I

(4.8.2)

then the observability grammian N1 (t0 , t0 + ν) of (C, A + KC > ) satisfies 0

0

β1 I ≤ N1 (t0 , t0 + ν) ≤ β2 I 0

0

(4.8.3)

for some constant β1 , β2 > 0. From the definition of the observability grammian matrix, (4.8.3) is equivalent to

222

CHAPTER 4. ON-LINE PARAMETER ESTIMATION Table 4.2 Gradient algorithms

Parametric model

z = θ∗> φ

Estimation model

zˆ = θ> φ

Normalized estimation error

z − zˆ m2 A. Based on instantaneous cost ²=

Adaptive law

θ˙ = Γ²φ

Design variables

m2 = 1 + n2s , n2s = αφ> φ, α > 0, Γ = Γ> > 0

Properties

(i) ², ²ns , θ, θ˙ ∈ L∞ ; (ii) ², ²ns , θ˙ ∈ L2 B. Based on the integral cost θ˙ = −Γ(Rθ + Q) > R˙ = −βR + φφ , R(0) = 0 m2 zφ ˙ Q = −βQ − m2 , Q(0) = 0

Adaptive law

Design variables

m2 = 1+n2s , ns chosen so that φ/m ∈ L∞ (e. g., n2s = αφ> φ, α > 0 ); β > 0, Γ = Γ> > 0

Properties

˙ R, Q ∈ L∞ ; (ii) ², ²ns , θ˙ ∈ L2 ; (i) ², ²ns , θ, θ, (iii) limt→∞ θ˙ = 0 Z 0

β1 |x1 (t0 )|2 ≤

t0 +ν

t0

0

|C > (t)x1 (t)|2 dt ≤ β2 |x1 (t0 )|2

(4.8.4)

where x1 is the state of the system x˙ 1 = (A + KC > )x1 y1 = C > x1 which is obtained, using output injection, from the system

Table 4.3 Least-squares algorithms

(4.8.5)

4.8. PARAMETER CONVERGENCE PROOFS Parametric model Estimation model

z = θ∗> φ zˆ = θ> φ

Normalized estimation error

² = (z − zˆ)/m2 A. Pure least-squares

Adaptive law Design variables

Properties

θ˙ = P ²φ > P, P (0) = P0 P˙ = −P φφ m2 > P0 = P0 > 0; m2 = 1 + n2s ns chosen so that φ/m ∈ L∞ (e.g., n2s = αφ> φ, α > 0 or n2s = φ> P φ) ˙ P ∈ L∞ ; (ii) ², ²ns , θ˙ ∈ L2 ; (iii) (i) ², ²ns , θ, θ, limt→∞ θ(t) = θ¯

B. Least-squares with covariance resetting θ˙ = P ²φ Adaptive law > P˙ = −P φφ P, P (t+ r ) = P0 = ρ0 I, m2 where tr is the time for which λmin (P ) ≤ ρ1 Design variables ρ0 > ρ1 > 0; m2 = 1 + n2s , ns chosen so that φ/m ∈ L∞ (e.g., n2s = αφ> φ, α > 0 ) ˙ P ∈ L∞ ; (ii) ², ²ns , θ˙ ∈ L2 (i) ², ²ns , θ, θ,

Properties

C. Least-squares with forgetting factor Adaptive law

Design variables Properties

θ˙ = P ²φ ( > if kP (t)k ≤ R0 βP − P φφ 2 P, ˙ m P = 0 otherwise P (0) = P0 m2 = 1 + n2s , n2s = αφ> φ or φ> P φ; β > 0, R0 > 0 scalars; P0 = P0> > 0, kP0 k ≤ R0 ˙ P ∈ L∞ ; (ii) ², ²ns , θ˙ ∈ L2 (i) ², ²ns , θ, θ,

223

224

CHAPTER 4. ON-LINE PARAMETER ESTIMATION Table 4.4 Adaptive laws for the bilinear model Parametric model : z = W (s)ρ∗ (θ∗> ψ + z0 ) A. SPR-Lyapunov design: sign of ρ∗ known

Parametric model rewritten Estimation model Normalized estimation error Adaptive law Design variables

Properties

z = W (s)L(s)ρ∗ (θ∗> φ + z1 ) φ = L−1 (s)ψ, z1 = L−1 (s)z0 zˆ = W (s)L(s)ρ(θ> φ + z1 ) ² = z − zˆ − W (s)L(s)²n2s θ˙ = Γ²φsgn(ρ∗ ) ρ˙ = γ²ξ, ξ = θ> φ + z1 L−1 (s) proper and stable; W (s)L(s) proper and φ z1 SPR; m2 = 1 + n2s ; ns chosen so that m , m ∈ L∞ > 2 > 2 (e.g. ns = α(φ φ+ z1 ), α > 0); Γ = Γ > 0, γ > 0 ˙ ρ˙ ∈ L2 (i) ², θ, ρ ∈ L∞ ; (ii) ², ²ns , θ,

B. Gradient algorithm: sign(ρ∗ ) known Parametric model z = ρ∗ (θ∗> φ + z1 ) rewritten φ = W (s)ψ, z1 = W (s)z0 Estimation model zˆ = ρ(θ> φ + z1 ) Normalized z − zˆ ²= m2 estimation error θ˙ = Γ²φsgn(ρ∗ ) Adaptive law ρ˙ = γ²ξ, ξ = θ> φ + z1 φ z1 Design variables m2 = 1 + n2s ; ns chosen so that m , m ∈ L∞ (e.g., 2 > 2 > ns = φ φ + z1 ); Γ = Γ > 0, γ > 0 Properties

˙ ρ˙ ∈ L∞ ; (ii) ², ²ns , θ, ˙ ρ˙ ∈ L2 (i) ², ²ns , θ, ρ, θ,

x˙ = Ax y = C >x 4

Form (4.8.5) and (4.8.6), it follows that e = x1 − x satisfies e˙ = Ae + KC > x1

(4.8.6)

4.8. PARAMETER CONVERGENCE PROOFS

225

Table 4.4 (Continued) C. Gradient algorithm with projection Sign (ρ∗ ) and lower bound 0 < ρ0 ≤ |ρ∗ | known z = θ¯∗> φ¯ Parametric model θ¯∗ = [θ¯1∗ , θ¯2∗> ]> , θ¯1∗ = ρ∗ , θ¯2∗ = ρ∗ θ∗ rewritten φ¯ = [z1 , φ> ]> Estimation model zˆ = θ¯> φ¯ z − zˆ Normalized ²= m2 estimation error  ∗ ¯  γ1 ²z1 if θ1 sgn(ρ ) > ρ0 or θ¯˙ 1 = if θ¯1 sgn(ρ∗ ) = ρ0 and −γ1 z1 ²sgn(ρ∗ ) ≤ 0   0 otherwise ˙θ = Γ ²φ ¯ 2 2 Adaptive law ¯ ρ = θ¯1 , θ = θθ¯2 1

Design variables

Properties

¯

φ ∈ L∞ (e.g., n2s = m2 = 1+n2s ; ns chosen so that m > ¯ α > 0 ); γ1 > 0; θ¯1 (0) satisfies |θ¯1 (0)| ≥ ρ0 ; αφ¯ φ, Γ2 = Γ > 2 > 0, γ > 0

˙ ρ˙ ∈ L∞ ; (ii) ², ²ns , θ, ˙ ρ˙ ∈ L2 (i) ², ²ns , θ, ρ, θ,

D. Gradient algorithm without projection Unknown sign (ρ∗ ) Parametric model z = ρ∗ (θ∗> φ + z1 ) zˆ = N (x)ρ(θ> φ + z1 ) N (x) = x2 cos x Estimation model ρ2 x = w + 2γ , w˙ = ²2 m2 , w(0) = 0 z − zˆ Normalized ²= m2 estimation error Adaptive law Design variables

Properties

θ˙ = N (x)Γ²φ ρ˙ = N (x)γ²ξ, ξ = θ> φ + z1 φ z1 m2 = 1 + n2s ; ns chosen so that m , m ∈ L∞ ; (e.g., > 2 > 2 ns = φ φ + z1 ); γ > 0, Γ = Γ > 0 ˙ ρ, ˙ ρ˙ ∈ L2 (i) ², ²ns , θ, ρ, θ, ˙ x, w ∈ L∞ ; (ii) ², ²ns , θ,

226

CHAPTER 4. ON-LINE PARAMETER ESTIMATION Table 4.5. Hybrid adaptive law

Parametric model

z = θ∗> φ

Estimation model

zˆ = θk> φ, t ∈ [tk , tk+1 )

Normalized estimation error

²=

Adaptive law

θk+1 = θk +Γ

Design variables

Sampling period Ts = tk+1 − tk > 0, tk = kTs ; m2 = 1 + n2s and ns chosen so that |φ|/m ≤ 1 (e.g., n2s = αφ> φ, α ≥ 1 ) Γ = Γ> > 0 2 − Ts λmax (Γ) > γ for some constant γ > 0

Properties

(i) θk ∈ l∞ , ², ²ns ∈ L∞ (ii) |θk+1 − θk | ∈ l2 ; ², ²ns ∈ L2

z − zˆ m2 R tk+1 tk

²(τ )φ(τ )dτ, k = 0, 1, 2, . . . ,

Consider the trajectories x(t) and x1 (t) with the same initial conditions. We have Z t e(t) = Φ(t, τ )K(τ )C > (τ )x1 (τ )dτ (4.8.7) t0

where Φ is the state transition matrix of (4.8.6). Defining ½ 4

x ¯1 =

KC > x1 /|KC > x1 | K/|K|

if |C > x1 | 6= 0 if |C > x1 | = 0

we obtain, using the Schwartz inequality, that Z |C > (t)e(t)|2 ≤ Z

t

¯ > ¯ ¯C (t)Φ(t, τ )K(τ )C > (τ )x1 (τ )¯2 dτ

t0 t

≤ t0

¯ > ¯2 ¯C (t)Φ(t, τ )¯ x1 (τ )¯ |K(τ )|2 dτ

Z

t t0

¯ > ¯ ¯C (τ )x1 (τ )¯2 dτ

(4.8.8)

4.8. PARAMETER CONVERGENCE PROOFS

227

Using the triangular inequality (a + b)2 ≤ 2a2 + 2b2 and (4.8.8), we have Z t0 +ν Z t0 +ν Z t0 +ν > 2 > 2 |C (t)x1 (t)| dt ≤ 2 |C (t)x(t)| dt + 2 |C > (t)e(t)|2 dt t0

≤

Z

t0 t0 +ν

2

|C > (t)x(t)|2 dt

t0

Z

t0 +ν

Z

t

+2 t0

≤

t0

¯ > ¯2 ¯C (t)Φ(t, τ )¯ x1 (τ )¯ |K(τ )|2 dτ

t0

Z

t

¯ > ¯ ¯C (τ )x1 (τ )¯2 dτ dt

t0

2β2 |x1 (t0 )|2 Z t0 +ν Z t Z t ¯ > ¯2 ¯ > ¯ ¯C (t)Φ(t, τ )¯ ¯C (τ )x1 (τ )¯2 dτ dt +2 x1 (τ )¯ |K(τ )|2 dτ t0

t0

t0

where the last inequality is obtained using the UCO property of (C, A) and the condition that x(t0 ) = x1 (t0 ). Applying the B-G Lemma, we obtain nR o Z t0 +ν t0 +ν R t 2 2|C > (t)Φ(t,τ )¯ x1 (τ )| |K(τ )|2 dτ dt > 2 2 t0 t0 |C (t)x1 (t)| dt ≤ 2β2 |x1 (t0 )| e (4.8.9) t0

By interchanging the sequence of integration, we have Z t0 +ν Z t Z t0 +ν Z t0 +ν ¯ > ¯2 ¯ > ¯2 ¯C (t)Φ(t, τ )¯ ¯C (t)Φ(t, τ )¯ x1 (τ )¯ |K(τ )|2 dτ dt = x1 (τ )¯ dt|K(τ )|2 dτ t0

t0

t0

τ

Because (C, A) being UCO and |¯ x1 | = 1 imply that Z t0 +ν Z t0 +ν ¯ > ¯2 ¯ > ¯ ¯C (t)Φ(t, τ )¯ ¯ ¯C (t)Φ(t, τ )¯2 dt ≤ β2 x1 (τ ) dt ≤ τ

t0

for any t0 ≤ τ ≤ t0 + ν, it follows from (4.8.1) and the above two equations that Z t0 +ν Z t ¯ > ¯2 ¯C (t)Φ(t, τ )¯ x1 (τ )¯ |K(τ )|2 dτ dt ≤ kν β2 (4.8.10) t0

t0

and, therefore, (4.8.9) leads to Z t0 +ν |C > (t)x1 (t)|2 dt ≤ 2β2 e2β2 kν |x1 (t0 )|2

(4.8.11)

t0

On the other hand, using x1 = x+e and the triangular inequality (a+b)2 ≥ 12 a2 −b2 , we have Z t0 +ν Z Z t0 +ν 1 t0 +ν > |C > (t)e(t)|2 dt |C > (t)x1 (t)|2 dt ≥ |C (t)x(t)|2 dt − 2 t0 t0 t0 Z t0 +ν β1 ≥ |x1 (t0 )|2 − |C > (t)e(t)|2 dt (4.8.12) 2 t0

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

where the last inequality is obtained using the UCO property of (C, A) and the fact x(t0 ) = x1 (t0 ). Substituting (4.8.8) for |C > e|2 , we obtain Z

t0 +ν

t0

β1 |x1 (t0 )|2 2 µZ t0 +ν ¶ Z t0 +ν Z t > 2 − |C (t)x1 (t)| dt |C > (t)Φ(t, τ )¯ x1 (τ )|2 |K(τ )|2 dτ dt |C > (t)x1 (t)|2 dt ≥

t0

t0

t0

by using the fact that Z

t

Z

t0 +ν

|C > (τ )x1 (τ )|2 dτ ≤

t0

|C > (τ )x1 (τ )|2 dτ

t0

for any t ≤ t0 + ν. Using (4.8.10), we have Z

t0 +ν

|C > (t)x1 (t)|2 dt ≥

t0

or

Z

t0 +ν

β1 |x1 (t0 )|2 − β2 kν 2

|C > (t)x1 (t)|2 dt ≥

t0 0

Setting β1 = 0

0

0 β1 2(1+β2 kν ) , β2

Z

t0 +ν

|C > (t)x1 (t)|2 dt

t0

β1 |x1 (t0 )|2 2(1 + β2 kν )

(4.8.13)

= 2β2 e2β2 kν , we have shown that (4.8.4) holds for

β1 , β2 > 0 and therefore (C, A + KC > ) is UCO, and, hence, the if part of the lemma is proved. The proof for the only if part is exactly the same as the if part since (C, A) can be obtained from (C, A + KC > ) using output injection. 2

Lemma 4.8.2 Let H(s) be a proper stable transfer function and y = H(s)u. If u ∈ L∞ , then Z t+T Z t+T 2 y (τ )dτ ≤ k1 u2 (τ )dτ + k2 t

t

for some constants k1 , k2 ≥ 0 and any t ≥ 0, T > 0. Furthermore, if H(s) is strictly proper, then k1 ≤ αkH(s)k2∞ for some constant α > 0. Proof Let us define

½ fS (τ ) =

f (τ ) if τ ∈ S 0 otherwise

where S ⊂ R is a subset of R that can be an open or closed interval. Then we can write u(τ ) = u[0,t) (τ ) + u[t,t+T ] (τ ), ∀0 ≤ τ ≤ t + T

4.8. PARAMETER CONVERGENCE PROOFS

229

Because H(s) can be decomposed as H(s) = h0 + H1 (s) where h0 is a constant and H1 (s) is strictly proper, we can express y = H(s)u as ª4 © y(τ ) = h0 u(τ )+H1 (s) u[0,t) + u[t,t+T ] = h0 u(τ )+y1 (τ )+y2 (τ ), ∀0 ≤ τ ≤ t+T 4

4

where y1 = H1 (s)u[0,t) , y2 = H1 (s)u[t,t+T ] . Therefore, using the inequality (a + b + c)2 ≤ 3a2 + 3b2 + 3c2 we have Z t+T

Z 2

y (τ )dτ ≤ t

3h20

Z

t+T

t+T

2

u (τ )dτ + 3 t

t

Z y12 (τ )dτ

t+T

+3 t

y22 (τ )dτ (4.8.14)

Using Lemma 3.3.1, Remark 3.3.2, and noting that y2 (τ ) = 0 for all τ < t, we have Z t+T Z t+T 4 y22 (τ )dτ = y22 (τ )dτ = ky2(t+T ) k22 ≤ kH1 (s)k2∞ ku[t,t+T ] k22 t

0

where

Z ku[t,t+T ] k22 =

Hence,

Z

Z

∞

u2[t,t+T ] (τ )dτ =

0

Z

t+T

y22 (τ )dτ ≤ kH1 (s)k2∞

t

t+T

u2 (τ )dτ

t

t+T

u2 (τ )dτ

(4.8.15)

t

To evaluate the second term in (4.8.14), we write Z τ y1 (τ ) = h1 (τ ) ∗ u[0,t) (τ ) = h1 (τ − σ)u[0,t) (σ)dσ 0 R ½ τ h (τ − σ)u(σ)dσ if τ < t R0t 1 = h (τ − σ)u(σ)dσ if τ ≥ t 0 1 where h1 (t) is the impulse response of H1 (s). Because H1 (s) is a strictly proper stable transfer function, |h1 (τ − σ)| ≤ α1 e−α2 (τ −σ) for some constants α1 , α2 > 0, then using the Schwartz inequality and the boundedness of u(τ ), we have ¶2 Z t+T Z t+T µZ t 2 y1 (τ )dτ = h1 (τ − σ)u(σ)dσ dτ t

t

Z

0

µZ

t+T

≤ ≤

t

|h1 (τ − σ)|dσ t

≤

Z

t

≤ α12

Z t

α12 α ¯ α22 α12 α ¯ 2α23

0

t+T

Z

µZ

0 t

Z

t

e−α2 (τ −σ) dσ

0 t+T

¶ |h1 (τ − σ)|u2 (σ)dσ dτ ¶ e−α2 (τ −σ) u2 (σ)dσ dτ

0

e−2α2 (τ −t) dτ

t

(4.8.16)

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

where α ¯ = supσ u2 (σ). Using (4.8.15) and (4.8.16) in (4.8.14) and defining k1 = 3α2 α ¯ 3(h20 + kH1 (s)k2∞ ), k2 = 2α13 , it follows that 2

Z

t+T

Z

t+T

2

y (τ )dτ ≤ k1 t

u2 (τ )dτ + k2

t

and the first part of the Lemma is proved. If H(s) is strictly proper, then h0 = 0, H1 (s) = H(s) and k1 = 3kH(s)k2∞ , therefore the proof is complete. 2 Lemma 4.8.3 (Properties of PE Signals) If w : R+ 7→ Rn is PE and w ∈ L∞ , then the following results hold: 4 (i) w1 = F w, where F ∈ Rm×n with m ≤ n is a constant matrix, is PE if and only if F is of rank m. (ii) Let (a) e ∈ L2 or (b) e ∈ L∞ and e → 0 as t → ∞, then w2 = w + e is PE. (iii) Let e ∈ S(µ) and e ∈ L∞ . There exists a µ∗ > 0 such that for all µ ∈ [0, µ∗ ), ωµ = ω + e is PE. (iv) If in addition w˙ ∈ L∞ and H(s) is a stable, minimum phase, proper rational transfer function, then w3 = H(s)w is PE. Proof The proof of (i) and (iii) is quite trivial, and is left as an exercise for the reader. To prove (iv), we need to establish that the inequality Z t+T β2 T ≥ (q > w3 (τ ))2 dτ ≥ β1 T (4.8.17) t

holds for some constants β1 , β2 , T > 0, any t ≥ 0 and any q ∈ Rn with |q| = 1. The existence of an upper bound in (4.8.17) is implied by the assumptions that w ∈ L∞ , H(s) has stable poles and therefore ω3 ∈ L∞ . To establish the lower bound, we define ar 4 z(t) = q> w (s + a)r and write ar ar > −1 z(t) = q H (s)w = H −1 (s)q > w3 3 (s + a)r (s + a)r where a > 0 is arbitrary at this moment and r > 0 is an integer that is chosen to be ar −1 equal to the relative degree of H(s) so that (s+a) (s) is a proper stable transfer rH function. According to Lemma 4.8.2, we have Z t+T Z t+T ¡ > ¢2 2 z (τ )dτ ≤ k1 q w3 (τ ) dτ + k2 t

t

4.8. PARAMETER CONVERGENCE PROOFS

231

for any t, T > 0 and some k1 , k2 > 0 which may depend on a, or equivalently ÃZ ! Z t+T t+T 1 > 2 2 (q w3 (τ )) dτ ≥ z (τ )dτ − k2 (4.8.18) k1 t t On the other hand, we have ar q> w (s + a)r µ r ¶ a − (s + a)r > q w+ q > w˙ s(s + a)r

z(t) = =

q > w + z1

= where

z1 =

ar − (s + a)r > q w˙ s(s + a)r

It is shown in Appendix A (see Lemma A.2) that ° r ° ° a − (s + a)r ° k ° ° ° s(s + a)r ° ≤ a ∞ for some constant k > 0 that is independent of a. Therefore, applying Lemma 4.8.2, we have Z t+T Z k3 t+T > 2 z12 (τ )dτ ≤ (q w) ˙ dτ + k¯3 a2 t t k4 T ≤ + k¯3 a2 where k3 = 3k 2 , k4 = k3 supt |ω| ˙ 2 and k¯3 ≥ 0 is independent of α, and the second inequality is obtained using the boundedness of w. ˙ Using the inequality (x + y)2 ≥ 1 2 2 x − y , we have 2 Z

t+T

Z z 2 (τ )dτ

t

Because w is PE, i.e.,

t+T

= t

≥

1 2

≥

1 2

Z t

t+T0

Z

(q > w(τ ) + z1 (τ ))2 dτ

t+T

Z (q > w)2 dτ −

t

Z

t

t t+T

t+T

z12 (τ )dτ

k4 T (q w) dτ − 2 − k¯3 a >

2

(q > w)2 dτ ≥ α0 T0

(4.8.19)

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

for some T0 , α0 > 0 and ∀t ≥ 0, we can divide the interval [t, t + T ] into subintervals of length T0 and write Z t+T n0 Z t+iT0 X (q > w)2 dτ ≥ (q > w)2 dτ ≥ n0 α0 T0 t

i=1

t+(i−1)T0

where n0 is the largest integer that satisfies n0 ≤ TT0 . From the definition of n0 , we have n0 ≥ TT0 − 1; therefore, we can establish the following inequality Z

t+T

µ >

2

(q w) dτ ≥ α0 t

¶ T − 1 T0 = α0 (T − T0 ) T0

(4.8.20)

Because the above analysis holds for any T > 0, we assume that T > T0 . Using (4.8.19) and (4.8.20) in (4.8.18), we have µ ¶ Z t+T 1 α0 k4 > 2 ¯ (q w3 ) dτ ≥ (T − T0 ) − 2 T − k3 − k2 k1 2 a t ½µ ¶ µ ¶¾ 1 α0 k4 α0 T0 ¯ − 2 T− + k3 + k2 (4.8.21) = k1 2 a 2 Since (4.8.21) holds for any a, T > 0 and α0 , k4 are independent of a, we can first choose a to satisfy α0 k4 α0 − 2 ≥ 2 a 4 and then fix T so that µ ¶ α0 α0 T0 ¯ T− + k3 + k2 > β1 k1 T 4 2 for a fixed β1 > 0. It follows that Z t+T (q > w3 (τ ))2 dτ ≥ β1 T > 0 t

and the lower bound in (4.8.17) is established.

2

Lemma 4.8.4 Consider the system Y˙ 1 = Ac Y1 − Bc φ> Y2 Y˙ 2 = 0 y0 = Cc> Y1

(4.8.22)

where Ac is a stable matrix, (Cc , Ac ) is observable, and φ ∈ L∞ . If φf defined as 4

φf = Cc> (sI − Ac )−1 Bc φ

4.8. PARAMETER CONVERGENCE PROOFS satisfies 1 T0

α1 I ≤

Z

t+T0 t

233

φf (τ )φ> f (τ )dτ ≤ α2 I, ∀t ≥ 0

(4.8.23)

for some constants α1 , α2 , T0 > 0, then (4.8.22) is UCO. Proof The UCO of (4.8.22) follows if we establish that the observability grammian N (t, t + T ) of (4.8.22) defined as 4

Z

t+T

N (t, t + T ) =

Φ> (τ, t)CC > Φ(τ, t)dτ

t

where C = [Cc> 0]> satisfies βI ≥ N (t, t + T ) ≥ αI for some constant α, β > 0, where Φ(t, t0 ) is the state transition matrix of (4.8.22). The upper bound βI follows from the boundedness of Φ(t, t0 ) that is implied by φ ∈ L∞ and the fact that Ac is a stable matrix. The lower bound will follow if we establish the following inequality: Z

t+T

t

¡ ¢ y02 (τ )dτ ≥ α |Y1 (t)|2 + |Y2 |2

where Y2 is independent of t due to Y˙ 2 = 0. From (4.8.22), we can write Z τ > > Ac (τ −t) y0 (τ ) = Cc Y1 (τ ) = Cc e Y1 (t) − Cc> eAc (τ −σ) Bc φ> (σ)dσY2 t

4

= y1 (τ ) + y2 (τ ) 4

4

for all τ ≥ t, where y1 (τ ) = Cc> eAc (τ −t) Y1 (t), y2 (τ ) = − x2 2

2

t

Cc> eAc (τ −σ) Bc φ> (σ)dσY2 .

y2 2

Using the inequalities (x + y) ≥ − y and (x + y) ≥ − x2 with x = y1 , y = y2 0 0 over the intervals [t, t + T ], [t + T , t + T ], respectively, we have Z

t+T t

2

Rτ

Z y02 (τ )dτ

t+T

≥ t

+

Z

2

0

y12 (τ ) dτ − 2

t+T

t+T 0

Z

y22 (τ ) dτ − 2

t+T

0

t

Z

0

t+T

t+T 0

y22 (τ )dτ y12 (τ )dτ

(4.8.24)

for any 0 < T < T . We now evaluate each term on the right-hand side of (4.8.24). Because Ac is a stable matrix, it follows that |y1 (τ )| ≤ k1 e−γ1 (τ −t) |Y1 (t)|

234

CHAPTER 4. ON-LINE PARAMETER ESTIMATION

for some k1 , γ1 > 0, and, therefore, Z t+T 0 k2 y12 (τ )dτ ≤ 1 e−2γ1 T |Y1 (t)|2 0 2γ1 t+T

(4.8.25)

On the other hand, since (Cc , Ac ) is observable, we have Z

t+T

0 >

eAc (t−τ ) Cc Cc> eAc (t−τ ) dτ ≥ k2 I

t 0

for any T > T1 and some constants k2 , T1 > 0. Hence, Z

t+T

0

t

y12 (τ )dτ ≥ k2 |Y1 (t)|2

(4.8.26)

Using y2 (τ ) = −φ> f (τ )Y2 and the fact that Z α2 n1 T0 I ≥

t+T

t+T 0

φf φ> f dτ ≥ n0 α1 T0 I

where n0 , n1 is the largest and smallest integer respectively that satisfy 0

T −T n0 ≤ ≤ n1 T0 0

i.e., n0 ≥ T −T T0 − 1, n1 ≤ isfied by y2 : Z

t+T

0

+ 1, we can establish the following inequalities satÃ

0

t

Z

T −T T0

t+T

t+T 0

y22 (τ )dτ

≤

y22 (τ )dτ

≥

! 0 T α2 T0 + 1 |Y2 |2 T0 Ã ! 0 T −T α1 T0 − 1 |Y2 |2 T0

Using (4.8.25), (4.8.26), (4.8.27) in (4.8.24), we have µ ¶ Z t+T 0 k2 k2 y02 (τ )dτ ≥ − 1 e−2γ1 T |Y1 (t)|2 2 2γ1 t Ã Ã ! Ã 0 !! 0 α1 T0 T − T T + − 1 − α2 T0 +1 |Y2 |2 2 T0 T0 0

0

(4.8.27)

(4.8.28)

Because the inequality (4.8.28) is satisfied for all T, T with T > T1 , let us first 0 0 choose T such that T > T1 and 0 k2 k2 k2 − 1 e−2γ1 T ≥ 2 2γ1 4

4.8. PARAMETER CONVERGENCE PROOFS

235

Now choose T to satisfy Ã

α1 T0 2

0

!

T −T −1 T0

Ã − α2 T0

0

T +1 T0

! ≥ β1

for a fixed β1 . We then have Z t+T ¡ ¢ y02 (τ )dτ ≥ α |Y1 (t)|2 + |Y2 (t)|2 t

where α = min

4.8.2

©

β1 , k42

ª

. Hence, (4.8.22) is UCO.

2

Proof of Corollary 4.3.1

Consider equations (4.3.30), (4.3.35), i.e., e˙ = Ac e + Bc (−θ˜> φ − ²n2s ) ˙ θ˜ = Γφ² ² = Cc> e

(4.8.29)

that describe the stability properties of the adaptive law. In proving Theorem 4.3.1, we have also shown that the time derivative V˙ of V =

e> Pc e θ˜> Γ−1 θ˜ + 2 2

where Γ = Γ> > 0 and Pc = Pc> > 0, satisfies V˙ ≤ −ν 0 ²2

(4.8.30)

0

for some constant ν > 0. Defining · ¸ Ac − Bc Cc> n2s −Bc φ> A(t) = , ΓφCc> 0

C = [Cc> 0]> , P =

1 2

·

Pc 0

and x = [e> , θ˜> ]> , we rewrite (4.8.29) as x˙ = A(t)x, ² = C > x and express the above Lyapunov-like function V and its derivative V˙ as V V˙

= x> P x = 2x> P Ax + x> P˙ x 0

0

= x> (P A + A> P + P˙ )x ≤ −ν x> CC > x = −ν ²2

0 Γ−1

¸

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

where P˙ = 0. It therefore follows that P , as defined above, satisfies the inequality 0

A> (t)P + P A(t) + ν CC > ≤ O Using Theorem 3.4.8, we can establish that the equilibrium ee = 0, θ˜e = 0 (i.e., xe = 0) of (4.8.29) is u.a.s, equivalently e.s., provided (C, A) is a UCO pair. According to Lemma 4.8.1, (C, A) and (C, A + KC > ) have the same UCO property, where · ¸ 4 Bc n2s K= −Γφ is bounded. We can therefore establish that (4.8.29) is UCO by showing that (C, A + KC > ) is a UCO pair. We write the system corresponding to (C, A + KC > ) as Y˙ 1 = Ac Y1 − Bc φ> Y2 (4.8.31) Y˙ 2 = 0 y0 = Cc> Y1 Because φ is PE and Cc> (sI − Ac )−1 Bc is stable and minimum phase (which is implied by Cc> (sI − Ac )−1 Bc being SPR) and φ˙ ∈ L∞ , it follows from Lemma 4.8.3 (iii) that Z τ

4

φf (τ ) = t

Cc> eAc (τ −σ) Bc φ(σ)dσ

is also PE; therefore, there exist constants α1 , α2 , T0 > 0 such that Z t+T0 1 α2 I ≥ φf (τ )φ> f (τ )dτ ≥ α1 I, ∀t ≥ 0 T0 t Hence, applying Lemma 4.8.4 to the system (4.8.31), we conclude that (C, A + KC > ) is UCO which implies that (C, A) is UCO. Therefore, we conclude that the equilibrium θ˜e = 0, ee = 0 of (4.8.29) is e.s. in the large. 2

4.8.3

Proof of Theorem 4.3.2 (iii)

The parameter error equation (4.3.53) may be written as ˙ θ˜ = A(t)θ˜ y0 = C > (t)θ˜ >

>

(4.8.32)

φ > where A(t) = −Γ φφ m2 , C (t) = − m , y0 = ²m. The system (4.8.32) is analyzed using the Lyapunov-like function

V =

θ˜> Γ−1 θ˜ 2

4.8. PARAMETER CONVERGENCE PROOFS

237

that led to (θ˜> φ)2 V˙ = − = −²2 m2 m2 along the solution of (4.8.32). We need to establish that the equilibrium θ˜e = 0 of (4.8.32) is e.s. We achieve that by using Theorem 3.4.8 as follows: Let P = Γ−1 , ˜> ˜ then V = θ 2P θ and 1 V˙ = θ˜> [P A(t) + A> (t)P + P˙ ]θ˜ = −θ˜> C(t)C > (t)θ˜ 2 where P˙ = 0. This implies that P˙ + P A(t) + A> P + 2C(t)C > (t) ≤ 0 and according to Theorem 3.4.8, θ˜e = 0 is e.s. provided (C, A) is UCO. Using Lemma 4.8.1, we have that (C, A) is UCO if (C, A + KC > ) is UCO for some K that satisfies the condition of Lemma 4.8.1. We choose K = −Γ

φ m

leading to A + KC > = 0. We consider the following system that corresponds to the pair (C, A + KC > ), i.e., Y˙ = 0 (4.8.33) > y0 = C > Y = − φm Y The observability grammian of (4.8.33) is given by Z N (t, t + T ) = t

t+T

φ(τ )φ> (τ ) dτ m2 (τ )

Because φ is PE and m ≥ 1 is bounded, it follows immediately that the grammian matrix N (t, t+T ) is positive definite for some T > 0 and for all t ≥ 0, which implies that (4.8.33) is UCO which in turn implies that (C, A) is UCO; thus, the proof is complete. 2 In the following, we give an alternative proof of Theorem 4.3.2 (iii), which does not make use of the UCO property. From (4.3.54), we have Z

t+T

V (t + T ) = V (t) −

Z 2

2

² m dτ = V (t) − t

t

t+T

(θ˜> (τ )φ(τ ))2 dτ m2

(4.8.34)

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

˜ )− for any t, T > 0. Expressing θ˜> (τ )φ(τ ) as θ˜> (τ )φ(τ ) = θ˜> (t)φ(τ ) + (θ(τ ˜ > φ(τ ) and using the inequality (x + y)2 ≥ 1 x2 − y 2 , it follows that θ(t)) 2

Z

t+T

t

≥

Z t+T ³ ´2 (θ˜> (τ )φ(τ ))2 1 ˜> (τ )φ(τ ) dτ dτ ≥ θ m2 m0 t ( Z t+T 1 1 (θ˜> (t)φ(τ ))2 dτ m0 2 t ) Z t+T ³ ´2 > ˜ ) − θ(t)) ˜ − (θ(τ φ(τ ) dτ

(4.8.35)

t

where m0 = supt≥0 m2 (t) is a constant. Because φ is PE, i.e., Z t+T0 φ(τ )φ> (τ )dτ ≥ α0 T0 I t

for some T0 and α0 > 0, we have Z t+T0 ³ ´2 θ˜> (t)φ(τ ) dτ

≥

˜ α0 T0 θ˜> (t)θ(t)

≥ =

−1 2α0 T0 λ−1 )V (t) max (Γ 2α0 T0 λmin (Γ)V (t)

t

On the other hand, we can write Z τ Z ˜ ) − θ(t) ˜ = ˜˙ θ(τ θ(σ)dσ =− t

t

τ

Γ

(4.8.36)

θ˜> (σ)φ(σ) φ(σ)dσ m2 (σ)

and Z ˜ ) − θ(t)) ˜ > φ(τ ) = (θ(τ

Ã

− Z

=

τ t

θ˜> (σ)φ(σ)Γφ(σ) m2

!> φ(τ )dσ

θ˜> (σ)φ(σ) φ> (τ )Γφ(σ) dσ m(σ) m(σ)

τ

− t

(4.8.37)

Noting that m ≥ 1, it follows from (4.8.37) and the Schwartz inequality that Z t+T0 ³ ´2 ˜ ) − θ(t)) ˜ > φ(τ ) dτ (θ(τ t  !2  ¶2 Z τ Ã ˜> Z t+T0 Z τ µ > φ (τ )Γφ(σ) θ (σ)φ(σ)  ≤ dσ dσ  dτ m(σ) m(σ) t t t Z ≤

β 4 λ2max (Γ)

Z

t+T0

(τ − t) t

t

τ

Ã

θ˜> (σ)φ(σ) m(σ)

!2 dσdτ

4.8. PARAMETER CONVERGENCE PROOFS

239

where β = supτ ≥0 |φ(τ )|. Changing the sequence of integration, we have Z

t+T0

³

´2 ˜ ) − θ(t)) ˜ > φ(τ ) dτ (θ(τ

t

Z ≤

β 4 λ2max (Γ)

≤

Ã

t

θ˜> (σ)φ(σ) m(σ)

!2 Z

t+T0

(τ − t)dτ dσ σ

½ ¾ (θ˜> (σ)φ(σ))2 T02 − (σ − t)2 dσ m2 (σ) 2 t Z β 4 λ2max (Γ)T02 t+T0 (θ˜> (σ)φ(σ))2 dσ 2 m2 (σ) t Z

≤

t+T0

β 4 λ2max (Γ)

t+T0

(4.8.38)

Using (4.8.36) and (4.8.38) in (4.8.35) with T = T0 we have Z t+T0 ˜> α0 T0 λmin (Γ) (θ (τ )φ(τ ))2 dτ ≥ V (t) 2 m (τ ) m0 t Z β 4 T02 λ2max (Γ) t+T0 (θ˜> (σ)φ(σ))2 − dσ 2m0 m2 (σ) t which implies Z t+T0 ˜> (θ (τ )φ(τ ))2 dτ m2 (τ ) t

≥ 4

= where γ1 =

2α0 T0 λmin (Γ) . 2m0 +β 4 T02 λ2max (Γ)

1 1+

β 4 T02 λ2max (Γ) 2m0

α0 T0 λmin (Γ) V (t) m0

γ1 V (t)

(4.8.39)

Using (4.8.39) in (4.8.34) with T = T0 , it follows that

V (t + T0 ) ≤ V (t) − γ1 V (t) = γV (t)

(4.8.40)

where γ = 1 − γ1 . Because γ1 > 0 and V (t + T0 ) ≥ 0, we have 0 < γ < 1. Because (4.8.40) holds for all t ≥ 0, we can take t = (n − 1)T0 and use (4.8.40) successively to obtain V (t) ≤ V (nT0 ) ≤ γ n V (0), ∀t ≥ nT0 , n = 0, 1, . . . ˜ → 0 as t → ∞ Hence, V (t) → 0 as t → ∞ exponentially fast which implies that θ(t) exponentially fast. 2

4.8.4

Proof of Theorem 4.3.3 (iv)

In proving Theorem 4.3.3 (i) to (iii), we have shown that (see equation (4.3.64)) ˜ V˙ (t) = −θ˜> (t)R(t)θ(t)

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

where V =

θ˜> Γ−1 θ˜ . 2

From equation (4.3.60), we have Z R(t) =

t

e−β(t−τ )

0

φ(τ )φ> (τ ) dτ m2 (τ )

Because φ is PE and m is bounded, we have Z

t

R(t) =

e−β(t−τ )

t−T0

Z

0

≥ α0 e−βT0

t

φ(τ )φ> (τ ) dτ + m2 (τ )

Z

t−T0

e−β(t−τ )

0

φ(τ )φ> (τ ) dτ m2 (τ )

φ(τ )φ> (τ )dτ

t−T0

≥ β1 e−βT0 I 0

0

for any t ≥ T0 , where β1 = α0 α0 T0 , α0 = supt m21(t) and α0 , T0 > 0 are constants given by (4.3.40) in the definition of PE. Therefore, V˙ ≤ −β1 e−βT0 θ˜> θ˜ ≤ −2β1 λmin (Γ)e−βT0 V for t ≥ T0 , which implies that V (t) satisfies V (t) ≤ e−α(t−T0 ) V (T0 ),

t ≥ T0

where α = 2β1 e−βT0 λmin 0 as t → ∞ exponentially fast with a p(Γ). Thus, V (t) → p ˜ ≤ 2λmax (Γ)V , we have that rate equal to α. Using 2λmin (Γ)V ≤ |θ| ˜ |θ(t)| ≤

s

p

2λmax (Γ)V (T0 )e

−α 2 (t−T0 )

≤

α λmax (Γ) ˜ |θ(T0 )|e− 2 (t−T0 ) , λmin (Γ)

t ≥ T0

Thus, θ(t) → θ∗ exponentially fast with a rate of α2 as t → ∞. Furthermore, for Γ = γI, α = 2β1 γe−βT0 and the rate of convergence (α/2) can be made large by increasing the value of the adaptive gain. 2

4.8.5

Proof of Theorem 4.3.4 (iv)

˜ satisfies the following In proving Theorem 4.3.4 (i) to (iii), we have shown that θ(t) equation ˜ = P (t)P −1 θ(0) ˜ θ(t) 0 We now show that P (t) → 0 as t → ∞ when φ satisfies the PE assumption. Because P −1 satisfies φφ> d −1 P = dt m2

4.8. PARAMETER CONVERGENCE PROOFS

241

R t+T using the condition that φ is PE, i.e., t 0 φ(τ )φ> (τ )dτ ≥ α0 T0 I for some constant α0 , T0 > 0, it follows that µ ¶ Z t φφ> α0 T0 t α0 T0 I −1 −1 P (t) − P (0) = dτ ≥ n0 I≥ −1 2 m m ¯ T m ¯ 0 0 where m ¯ = supt {m2 (t)} and n0 is the largest integer that satisfies n0 ≤ n0 ≥ Tt0 − 1. Therefore, ¶ µ α0 T0 t −1 −1 −1 I P (t) ≥ P (0) + T0 m ¯ µ ¶ t α0 T0 ≥ −1 I, ∀t > T0 T0 m ¯ which, in turn, implies that µµ ¶ ¶−1 t P (t) ≤ − 1 α0 T0 mI, ¯ T0

∀t > T0

t T0 ,

i.e.,

(4.8.41)

Because P (t) ≥ 0 for all t ≥ 0 and the right-hand side of (4.8.41) goes to zero as ˜ = P (t)P −1 θ(0) ˜ →0 t → ∞, we can conclude that P (t) → 0 as t → ∞. Hence, θ(t) 0 as t → ∞. 2

4.8.6

Proof of Corollary 4.3.2

Let us denote Γ = P −1 (t), then from (4.3.78) we have φφ> Γ˙ = −βΓ + 2 , m or

Γ(0) = Γ> (0) = Γ0 = P0−1 Z

Γ(t) = e−βt Γ0 +

t

e−β(t−τ )

0

φ(τ )φ> (τ ) dτ m2

Using the condition that φ(t) is PE and m ∈ L∞ , we can show that for all t ≥ T0 Z t φφ> Γ(t) ≥ e−β(t−τ ) 2 dτ m 0 Z t Z t−T0 φ(τ )φ> (τ ) φ(τ )φ> (τ ) dτ + e−β(t−τ ) dτ = e−β(t−τ ) 2 m m2 0 t−T0 α0 T0 ≥ e−βT0 I (4.8.42) m ¯ where m ¯ = supt m2 (t). For t ≤ T0 , we have Γ(t) ≥ e−βt Γ0 ≥ e−βT0 Γ0 ≥ λmin (Γ0 )e−βT0 I

(4.8.43)

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

Conditions (4.8.42), (4.8.43) imply that Γ(t) ≥ γ1 I

(4.8.44)

for all t ≥ 0 where γ1 = min{ αβ0 T1 0 , λmin (Γ0 )}e−βT0 . On the other hand, using the boundedness of φ, we can establish that for some constant β2 > 0 Z t e−β(t−τ ) dτ I Γ(t) ≤ Γ0 + β2 0

≤

λmax (Γ0 )I +

β2 I ≤ γ2 I β

(4.8.45)

where γ2 = λmax (Γ0 ) + ββ2 > 0. Combining (4.8.44) and (4.8.45), we conclude γ1 I ≤ Γ(t) ≤ γ2 I for some γ1 > 0, γ2 > 0. Therefore, γ2−1 I ≤ P (t) ≤ γ1−1 I and consequently P (t), P −1 (t) ∈ L∞ . Because P (t), P −1 (t) ∈ L∞ , the exponential convergence of θ to θ∗ can be proved using exactly the same procedure and arguments as in the proof of Theorem 4.3.2. 2

4.8.7

Proof of Theorem 4.5.1(iii)

Consider the following differential equations which describe the behavior of the adaptive law (see (4.5.6) and (4.5.7)): e˙ = Ac e + Bc (−ρ∗ θ˜> φ − ρ˜ξ − ²n2s ) ˙ θ˜ = Γ²φsgn(ρ∗ ) ρ˙ = γ²ξ ² = Cc> e

(4.8.46)

Because ξ, ρ ∈ L∞ and ξ ∈ L2 , we can treat ξ, ρ as external input functions and write (4.8.46) as x˙ a = Aa xa + Ba (−˜ ρξ) (4.8.47) where · xa =

e θ˜

¸

· ,

Aa =

Ac − n2s Bc Cc> Γsgn(ρ∗ )φCc>

−ρ∗ Bc φ> 0

¸

· ,

Ba =

Bc 0

¸

4.8. PARAMETER CONVERGENCE PROOFS

243

In proving Corollary 4.3.1, we have shown that when φ is PE and φ, φ˙ ∈ L∞ , the system x˙ = Aa x is e.s. Therefore, the state transition matrix Φa (t, t0 ) of (4.8.47) satisfies kΦa (t, 0)k ≤ α0 e−γ0 t (4.8.48) for some constants α0 , γ0 > 0, which together with −Ba ρ˜ξ ∈ L2 imply that xa (t) → 0 as t → ∞.

4.8.8

Proof of Theorem 4.6.1 (iii)

From the proof of Theorem 4.6.1 (i) to (ii), we have the inequality (see (4.6.9)) Z tk+1 V (k + 1) − V (k) ≤ −(2 − Ts λm ) ²2 (τ )m2 (τ )dτ (4.8.49) tk

Using inequality (4.8.49) consecutively, we have Z V (k + n) − V (k) ≤

tk+n

−(2 − Ts λm )

²2 (τ )m2 (τ )dτ

tk

n−1 X Z tk+i+1

²2 (τ )m2 (τ )dτ

= −(2 − Ts λm )

i=0

(4.8.50)

tk+i

for any integer n. We now write Z

tk+i+1

Z ²2 (τ )m2 (τ )dτ

³ tk+i+1

=

tk+i

tk+i

Z

´2 > θ˜k+i φ(τ ) m2 (τ )

³

tk+i+1

=

dτ

´2 θ˜k> φ(τ ) + (θ˜k+i − θ˜k )> φ(τ ) m2 (τ )

tk+i

dτ

Using the inequality (x + y)2 ≥ 12 x2 − y 2 , we write ³ ´2 ³ ´2 Z tk+i+1 Z tk+i+1 θ˜> φ(τ ) Z tk+i+1 (θ˜k+i − θ˜k )> φ(τ ) k 1 ²2 (τ )m2 (τ )dτ ≥ dτ − dτ 2 tk+i m2 (τ ) m2 (τ ) tk+i tk+i (4.8.51) |φ(τ )|2 Because φ(τ )/m(τ ) is bounded, we denote c = sup m2 (τ ) and have Z

³ tk+i+1

tk+i

(θ˜k+i − θ˜k )> φ(τ ) m2 (τ )

´2 dτ ≤ cTs |θ˜k+i − θ˜k |2

From the hybrid adaptive algorithm, we have Z tk+i ˜ ˜ θk+i − θk = ²(τ )φ(τ )dτ, i = 1, 2, . . . , n tk

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

therefore, using the Schwartz inequality and the boundedness of |φ(t)|/m(t), µZ tk+i ¶2 µZ tk+i ¶2 |φ(τ )| |θ˜k+i − θ˜k |2 ≤ |²(τ )||φ(τ )|dτ = |²(τ )|m(τ ) dτ m(τ ) tk tk Z tk+i Z tk+i 2 |φ(τ )| ≤ ²2 (τ )m2 (τ )dτ dτ m2 (τ ) tk tk Z tk+i ≤ ciTs ²2 (τ )m2 (τ )dτ tk tk+n

Z ≤

ciTs

²2 (τ )m2 (τ )dτ

(4.8.52)

tk

Using the expression (4.8.52) in (4.8.51), we have ³ ´2 Z tk+i+1 Z tk+i+1 θ˜> φ(τ ) Z tk+n k 1 2 2 2 2 ² (τ )m (τ )dτ ≥ dτ − c iTs ²2 (τ )m2 (τ )dτ 2 tk+i m2 (τ ) tk+i tk which leads to Z tk+n n−1 XZ ²2 (τ )m2 (τ )dτ = tk

≥

≥ =

 n−1 X

1  2 i=0

i=0

Z

³ tk+i+1

tk+i

²2 (τ )m2 (τ )dτ

tk+i

θ˜k> φ(τ )

´2

m2 (τ )



Z dτ − c2 iTs2

tk+n

 ²2 (τ )m2 (τ )dτ 

tk

Z tk+n n−1 Z n−1 X 1 ˜> X tk+i+1 φ(τ )φ> (τ ) ˜ 2 2 θk dτ θ − c iT ²2 (τ )m2 (τ )dτ k s 2 (τ ) 2 m tk i=0 tk+i i=0 Z tk+n Z tk+n > n(n − 1) 2 2 φ(τ )φ (τ ) ˜ dτ θk − c Ts ²2 (τ )m2 (τ )dτ θ˜k> 2 (τ ) 2m 2 tk tk

or equivalently Z tk+n ²2 (τ )m2 (τ )dτ ≥ tk

tk+i+1

1 θ˜> 2(1 + n(n − 1)c2 Ts2 /2) k

Z

tk+n

tk 0

φ(τ )φ> (τ ) ˜ dτ θk (4.8.53) m2 (τ ) 0

Because φ is PE and 1 ≤ m < ∞, there exist constants α1 , α2 and T0 > 0 such that Z t+T0 0 0 φ(τ )φ> (τ ) α2 I ≥ dτ ≥ α1 I 2 m t for any t. Therefore, for any integer k, n where n satisfies nTs ≥ T0 , we have Z tk+n 0 V (k) 0 φ(τ )φ> (τ ) ˜ > ˜ dτ θk ≥ α1 θ˜k> θ˜k ≥ α (4.8.54) θk 2 m λm 1 tk

4.9. PROBLEMS

245

Using (4.8.53), (4.8.54) in (4.8.50), we obtain the following inequality: 0

V (k + n) − V (k) ≤ −

(2 − Ts λm )α1 V (k) λm (2 + n(n − 1)c2 Ts2 )

(4.8.55)

hold for any integer n with n ≥ T0 /Ts . Condition (4.8.55) is equivalent to V (k + n) ≤ γV (k) with

0

4

γ =1−

(2 − Ts λm )α1 <1 λm (2 + n(n − 1)c2 Ts2 )

Therefore, V (kn) = V ((k−)n + n) ≤ γV ((k − 1)n) ≤ γ 2 V ((k − 2)n) ≤ . . . ≤ γ k V (0) or

s

s V (kn) ≤ λm

|θ˜kn | ≤

V (0) √ k ( γ) λm

(4.8.56)

√ Because 0 < γ < 1 and, therefore, γ < 1, (4.8.56) implies that |θ˜kn | → 0 exponentially fast as t → ∞, which, together with the property of the hybrid adaptive algorithm (i.e., |θ˜k+1 | ≤ |θ˜k |), implies that θk converges to θ∗ exponentially fast and the proof is complete.

4.9

Problems

4.1 Consider the differential equation ˙ θ˜ = −γu2 θ˜ given by (4.2.7) where γ > 0. Show that a necessary and sufficient condition ˜ to converge to zero exponentially fast is that u(t) satisfies (4.2.11), for θ(t) i.e., Z t+T0 u2 (τ )dτ ≥ α0 T0 t

for all t ≥ 0 and some constants α0 , T0 > 0. (Hint: Show that e

−γ

Rt t1

u2 (τ )dτ

≤ αe−γ0 (t−t1 )

for all t ≥ t1 and some α, γ0 > 0 if and only if (4.2.11) is satisfied.)

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

4.2 Consider the second-order stable system · ¸ · ¸ a11 a12 b1 x˙ = x+ u a21 0 b2 where x, u are available for measurement, u ∈ L∞ and a11 , a12 , a21 , b1 , b2 are unknown parameters. Design an on-line estimator to estimate the unknown parameters. Simulate your scheme using a11 = −0.25, a12 = 3, a21 = −5, b1 = 1, b2 = 2.2 and u = 10sin2t. Repeat the simulation when u = 10sin2t + 7cos3.6t. Comment on your results. 4.3 Consider the nonlinear system x˙ = a1 f1 (x) + a2 f2 (x) + b1 g1 (x)u + b2 g2 (x)u where u, x ∈ R1 , fi , gi are known nonlinear functions of x and ai , bi are unknown constant parameters. The system is such that u ∈ L∞ implies x ∈ L∞ . If x, u can be measured at each time, design an estimation scheme for estimating the unknown parameters on-line. 4.4 Design and analyze an on-line estimation scheme for estimating θ∗ in (4.3.26) when L(s) is chosen so that W (s)L(s) is biproper and SPR. 4.5 Design an on-line estimation scheme to estimate the coefficients of the numerator polynomial Z(s) = bn−1 sn−1 + bn−2 sn−2 + · · · + b1 s + b0 of the plant y=

Z(s) u R(s)

when the coefficients of R(s) = sn + an−1 sn−1 + · · · + a1 s + a0 are known. Repeat the same problem when Z(s) is known and R(s) is unknown. 4.6 Consider the cost function given by (4.3.51), i.e. J(θ) =

(z − θ> φ)2 2m2

which we like to minimize w.r.t. θ. Derive the nonrecursive algorithm θ(t) =

φz φ> φ

provided φ> φ 6= 0 by solving the equation ∇J(θ) = 0. 4.7 Show that ω0 = F ω, where F ∈ Rm×n with m ≤ n is a constant matrix and ω ∈ L∞ is PE, is PE if and only if F is of full rank.

4.9. PROBLEMS

247

4.8 Show that if ω, ω˙ ∈ L∞ , ω is PE and either (a) e ∈ L2 or (b) e ∈ L∞ and e(t) → 0 as t → ∞ is satisfied, then ω0 = ω + e is PE. 4.9 Consider the mass-spring-damper system shown in Figure 4.11. y(t) k u

m β

Figure 4.11 The mass-spring-damper system for Problem 4.9. where β is the damping coefficient, k is the spring constant, u is the external force, and y(t) is the displacement of the mass m resulting from the force u. (a) Verify that the equations of the motion that describe the dynamic behavior of the system under small displacements are m¨ y + β y˙ + ky = u (b) Design a gradient algorithm to estimate the constants m, β, k when y, u can be measured at each time t. (c) Repeat (b) for a least squares algorithm. (d) Simulate your algorithms in (b) and (c) on a digital computer by assum2 ing that m = 20 kg, β = 0.1 kg/sec, k = 5 kg/sec and inputs u of your choice. (e) Repeat (d) when m = 20 kg for 0 ≤ t ≤ 20 sec and m = 20(2 − e−0.01(t−20) ) kg for t ≥ 20sec. 4.10 Consider the mass-spring-damper system shown in Figure 4.12.

y2(t)

y1(t)

m β

k

u

Figure 4.12 The mass-spring-damper system for Problem 4.10.

248

CHAPTER 4. ON-LINE PARAMETER ESTIMATION (a) Verify that the equations of motion are given by k(y1 − y2 ) = u k(y1 − y2 ) = m¨ y2 + β y˙ 2 (b) If y1 , y2 , u can be measured at each time t, design an on-line parameter estimator to estimate the constants k, m and β. (c) We have the a priori knowledge that 0 ≤ β ≤ 1, k ≥ 0.1 and m ≥ 10. Modify your estimator in (b) to take advantage of this a priori knowledge. (d) Simulate your algorithm in (b) and (c) when β = 0.2 kg/sec, m = 15 kg, 2 k = 2 kg/sec and u = 5 sin 2t + 10.5 kg · m/sec2 .

4.11 Consider the block diagram of a steer-by-wire system of an automobile shown in Figure 4.13. r + l Σ − 6

θp

- G0 (s)

- G1 (s)

θ˙ -

θpFigure 4.13 Block diagram of a steer-by-wire system for Problem 4.11. where r is the steering command in degrees, θp is the pinion angle in degrees and θ˙ is the yaw rate in degree/sec. The transfer functions G0 (s), G1 (s) are of the form k0 ω02 G0 (s) = 2 s + 2ξ0 ω0 s + ω02 (1 − k0 ) G1 (s) =

s2

k1 ω12 + 2ξ1 ω1 s + ω12

where k0 , ω0 , ξ0 , k1 , ω1 , ξ1 are functions of the speed of the vehicle. Assuming that r, θp , θ˙ can be measured at each time t, do the following: (a) Design an on-line parameter estimator to estimate ki , ωi , ξi , i = 0, 1 using ˙ r. the measurement of θp , θ, (b) Consider the values of the parameters shown in Table 4.6 at different speeds: Table 4.6 Parameter values for the SBW system Speed V k0 ω0 ξ0 k1 ω1 ξ1 30 mph 0.81 19.75 0.31 0.064 14.0 0.365 60 mph 0.77 19.0 0.27 0.09 13.5 0.505

4.9. PROBLEMS

249

Assume that between speeds the parameters vary linearly. Use these values to simulate and test your algorithm in (a) when (i) r = 10 sin 0.2t + 8 degrees and V = 20 mph. (ii) r = 5 degrees and the vehicle speeds up from V = 30 mph to V = 60 mph in 40 second with constant acceleration and remains at 60 mph for 10 second. 4.12 Show that the hybrid adaptive law with projection presented in Section 4.6 guarantees the same properties as the hybrid adaptive law without projection. 4.13 Consider the equation of the motion of the mass-spring-damper system given in Problem 4.9, i.e., m¨ y + β y˙ + ky = u This system may be written in the form: y = ρ∗ (u − m¨ y − β y) ˙ where ρ∗ = k1 appears in a bilinear form with the other unknown parameters m, β. Use the adaptive law based on the bilinear parametric model to estimate ρ∗ , m, β when u, y are the only signals available for measurement. Because k > 0, the sign of ρ∗ may be assumed known. Simulate your adaptive law using the numerical values given in (d) and (e) of Problem 4.9.

Chapter 5

Parameter Identifiers and Adaptive Observers 5.1

Introduction

In Chapter 4, we developed a wide class of on-line parameter estimation schemes for estimating the unknown parameter vector θ∗ that appears in certain general linear and bilinear parametric models. As shown in Chapter 2, these models are parameterizations of LTI plants, as well as of some special classes of nonlinear plants. In this chapter we use the results of Chapter 4 to design parameter identifiers and adaptive observers for stable LTI plants. We define parameter identifiers as the on-line estimation schemes that guarantee convergence of the estimated parameters to the unknown parameter values. The design of such schemes includes the selection of the plant input so that a certain signal vector φ, which contains the I/O measurements, is PE. As shown in Chapter 4, the PE property of φ guarantees convergence of the estimated parameters to the unknown parameter values. A significant part of Section 5.2 is devoted to the characterization of the class of plant inputs that guarantee the PE property of φ. The rest of Section 5.2 is devoted to the design of parameter identifiers for plants with full and partial state measurements. In Section 5.3 we consider the design of schemes that simultaneously estimate the plant state variables and parameters by processing the plant I/O measurements on-line. We refer to such schemes as adaptive observers. 250

5.2. PARAMETER IDENTIFIERS

251

The design of an adaptive observer is based on the combination of a state observer that could be used to estimate the state variables of a particular plant state-space representation with an on-line estimation scheme. The choice of the plant state-space representation is crucial for the design and stability analysis of the adaptive observer. We present several different plant state-space representations that we then use to design and analyze stable adaptive observers in Sections 5.3 to 5.5. In Section 5.6, we present all the lengthy proofs dealing with parameter convergence. The stability properties of the adaptive observers developed are based on the assumption that the plant is stable and the plant input is bounded. This assumption is relaxed in Chapter 7, where adaptive observers are combined with control laws to form adaptive control schemes for unknown and possibly unstable plants.

5.2

Parameter Identifiers

Consider the LTI plant represented by the vector differential equation x˙ = Ax + Bu, y = C >x

x(0) = x0

(5.2.1)

where x ∈ Rn , u ∈ Rq , y ∈ Rm and n ≥ m. The elements of the plant input u are piecewise continuous, uniformly bounded functions of time. The plant parameters A ∈ Rn×n , B ∈ Rn×q , C ∈ Rn×m are unknown constant matrices, and A is a stable matrix. We assume that the plant (5.2.1) is completely controllable and completely observable i.e., (5.2.1) is a minimal state-space representation of the plant. The objective of this section is to design on-line parameter estimation schemes that guarantee convergence of the estimated plant parameters to the true ones. We refer to this class of schemes as parameter identifiers. The plant parameters to be estimated are the constant matrices A, B, C in (5.2.1) or any other set of parameters in an equivalent plant parameterization. The design of a parameter identifier consists of two steps. In the first step we design an adaptive law for estimating the parameters of a convenient parameterization of the plant (5.2.1) by following any one of the procedures developed in Chapter 4.

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CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

In the second step, we select the input u so that the adaptive law designed in the first step guarantees that the estimated parameters converge to the true ones. As shown in Chapter 4, parameter convergence requires additional conditions on a signal vector φ that are independent of the type of the adaptive law employed. In general the signal vector φ is related to the plant input or external input command u through the equation φ = H(s)u

(5.2.2)

where H(s) is some proper transfer matrix with stable poles. The objective of the second step is to choose the input u so that φ is P E, which, as shown in Chapter 4, guarantees convergence of the estimated parameters to their true values. Before we embark on the design of parameter identifiers, let us first characterize the class of input signals u that guarantee the P E property of the signal vector φ in (5.2.2). The relationship between φ, H(s) and the plant equation (5.2.1) will be explained in the sections to follow.

5.2.1

Sufficiently Rich Signals

Let us consider the first order plant y˙ = −ay + bu,

y(0) = y0

or

b u (5.2.3) s+a where a > 0, b are unknown constants and y, u ∈ L∞ are measured at each time t. We would like to estimate a, b by properly processing the input/output data y, u. It is clear that for the estimation of a, b to be possible, the input/output data should contain sufficient information about a, b. For example, for u = 0 the output y=

y(t) = e−at y0 carries no information about the parameter b. If, in addition, y0 = 0 then y(t) ≡ 0 ∀t ≥ 0 obviously carries no information about any one of the plant

5.2. PARAMETER IDENTIFIERS

253

parameters. If we now choose u = constant c0 6= 0, then ¶

µ

b b y(t) = e−at y0 − c0 + c0 a a carries sufficient information about a, b provided y0 6= ac0 /b. This information disappears exponentially fast, and at steady state y≈

b c0 a

carries information about the zero frequency gain ab of the plant only, which is not sufficient to determine a, b uniquely. Let us now choose u(t) = sin ω0 t for some ω0 > 0. The steady-state response of the plant (5.2.3) is given by y ≈ A sin(ω0 t + ϕ) where A=

|b| |b| ω0 =q , ϕ = (sgn(b) − 1)90◦ − tan−1 |jω0 + a| a ω02 + a2

(5.2.4)

It is clear that measurements of the amplitude A and phase ϕ uniquely determine a, b by solving (5.2.4) for the unknown a, b. The above example demonstrates that for the on-line estimation of a, b to be possible, the input signal has to be chosen so that y(t) carries sufficient information about a, b. This conclusion is obviously independent of the method or scheme used to estimate a, b. Let us now consider an on-line estimation scheme for the first-order plant (5.2.3). For simplicity we assume that y˙ is measured and write (5.2.3) in the familiar form of the linear parametric model z = θ∗> φ where z = y, ˙ θ∗ = [b, a]> , φ = [u, −y]> . Following the results of Chapter 4, we consider the following gradient algorithm for estimating θ∗ given in Table 4.2. θ˙ = Γ²φ, Γ = Γ> > 0 ² = z − zˆ, zˆ = θ> φ

(5.2.5)

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CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

where θ(t) is the estimate of θ∗ at time t and the normalizing signal m2 = 1 due to φ ∈ L∞ . T As we established in Chapter 4, (5.2.5) guarantees that ², θ˙ ∈ L2 L∞ , ˙ → 0 as t → ∞. If in addition φ is P E, θ ∈ L∞ , and if u˙ ∈ L∞ then ²(t), θ(t) i.e., it satisfies 1 α1 I ≥ T0

Z t+T0 t

φ(τ )φ> (τ )dτ ≥ α0 I,

∀t ≥ 0

(5.2.6)

for some T0 , α0 , α1 > 0, then θ(t) → θ∗ exponentially fast. Because the vector φ is given by " # 1 φ= u b − s+a the convergence of θ(t) to θ∗ is guaranteed if we choose u so that φ is P E, i.e., it satisfies (5.2.6). Let us try the choices of input u considered earlier. For u = c0 the vector φ at steady state is given by "

φ≈

c0 − bca0

#

which does not satisfy the right-hand-side inequality in (5.2.6) for any constant c0 . Hence, for u = c0 , φ cannot be P E and, therefore, θ(t) cannot be guaranteed to converge to θ∗ exponentially fast. This is not surprising since as we showed earlier, for u = c0 , y(t) does not carry sufficient information about a, b at steady state. On the other hand, for u = sin t we have that at steady state "

φ≈

sin t −A sin(t + ϕ)

#

|b| where A = √1+a , ϕ = (sgn(b) − 1)90◦ − tan−1 a1 , which can be shown to 2 satisfy (5.2.6). Therefore, for u = sin t the signal vector φ carries sufficient information about a and b, φ is P E and θ(t) → θ∗ exponentially fast. We say that u = sin t is sufficiently rich for identifying the plant (5.2.3), i.e., it contains a sufficient number of frequencies to excite all the modes of the plant. Because u = c0 6= 0 can excite only the zero frequency gain of the plant, it is not sufficiently rich for the plant (5.2.3).

5.2. PARAMETER IDENTIFIERS

255

Let us consider the second order plant y=

s2

b1 s + b0 u = G(s)u + a1 s + a0

(5.2.7)

where a1 , a0 > 0 and G(s) has no zero-pole cancellations. We can show that for u = sin ω0 t, y(t) at steady state does not carry sufficient information to be able to uniquely determine a1 , a0 , b1 , b0 . On the other hand, u(t) = sin ω0 t + sin ω1 t where ω0 6= ω1 leads to the steady-state response y(t) = A0 sin(ω0 t + ϕ0 ) + A1 sin(ω1 t + ϕ1 ) where A0 =| G(jω0 ) |, ϕ0 = 6 G(jω0 ), A1 =| G(jω1 ) |, ϕ1 = 6 G(jω1 ). By measuring A0 , A1 , ϕ0 , ϕ1 we can determine uniquely a1 , a0 , b1 , b0 by solving four algebraic equations. Because each frequency in u contributes two equations, we can argue that the number of frequencies that u should contain, in general, is proportional to the number of unknown plant parameters to be estimated. We are now in a position to give the following definition of sufficiently rich signals. Definition 5.2.1 A signal u : R+ → R is called sufficiently rich of order n if it consists of at least n2 distinct frequencies. For example, the input u=

m X

Ai sin ωi t

(5.2.8)

i=1

where m ≥ n/2, Ai 6= 0 are constants and ωi 6= ωk for i 6= k is sufficiently rich of order n. A more general definition of sufficient richness that includes signals that are not necessarily equal to a sum of sinusoids is presented in [201] and is given below. Definition 5.2.2 A signal u : R+ → Rn is said to be stationary if the following limit exists uniformly in t0 1 Ru (t) = lim T →∞ T

Z t0 +T t0

u(τ )u> (t + τ )dτ

256

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

The matrix Ru (t) ∈ Rn×n is called the autocovariance of u. Ru (t) is a positive semidefinite matrix and its Fourier transform given by Su (ω) =

Z ∞ −∞

e−jωτ Ru (τ )dτ

is referred to as the spectral measure of u. If u has a sinusoidal component at frequency ω0 then u is said to have a spectral line at frequency ω0 and Su (ω) has a point mass (a delta function) at ω0 and −ω0 . Given Su (ω), Ru (t) can be calculated using the inverse Fourier transform, i.e., Ru (t) =

1 2π

Z ∞ −∞

ejωt Su (ω)dω

Furthermore, we have Z ∞ −∞

Su (ω)dω = 2πRu (0)

For further details about the properties of Ru (t), Su (ω), the reader is referred to [186, 201]. Definition 5.2.3 A stationary signal u : R+ → R is called sufficiently rich of order n, if the support of the spectral measure Su (ω) of u contains at least n points. Definition 5.2.3 covers a wider class of signals that includes those specified by Definition 5.2.1. For example, the input (5.2.8) has a spectral measure with 2m points of support, i.e., at ωi , −ωi for i = 1, 2, . . . m, where m ≥ n/2, and is, therefore, sufficiently rich of order n. Let us now consider the equation φ = H(s)u

(5.2.9)

where H(s) is a proper transfer matrix with stable poles and φ ∈ Rn . The P E property of φ is related to the sufficient richness of u by the following theorem given in [201]. Theorem 5.2.1 Let u : R+ 7→ R be stationary and assume that H(jω1 ), . . ., H(jωn ) are linearly independent on C n for all ω1 , ω2 , . . . , ωn ∈ R, where ωi 6= ωk for i 6= k. Then φ is P E if, and only if, u is sufficiently rich of order n.

5.2. PARAMETER IDENTIFIERS

257

The proof of Theorem 5.2.1 is given in Section 5.6. The notion of persistence of excitation of the vector φ and richness of the input u attracted the interest of several researchers in the 1960s and 1970s who gave various interpretations to the properties of P E and sufficiently rich signals. The reader is referred to [1, 171, 201, 209, 242] for further information on the subject. Roughly speaking, if u has at least one distinct frequency component for each two unknown parameters, then it is sufficiently rich. For example, if the number of unknown parameters is n, then m ≥ n2 distinct frequencies in u are sufficient for u to qualify as being sufficiently rich of order n. Of course, these statements are valid provided H(jω1 ), . . . , H(jωn ) with ωi 6= ωk are linearly independent on C n for all ωi ∈ R, i = 1, 2, . . . , n. The vectors H(jωi ), i = 1, 2, . . . , n may become linearly dependent at some frequencies in R under certain conditions such as the one illustrated by the following example where zeros of the plant are part of the internal model of u. Example 5.2.1 Let us consider the following plant: y=

b0 (s2 + 1) u = G(s)u (s + 2)3

where b0 is the only unknown parameter. Following the procedure of Chapter 2, we rewrite the plant in the form of the linear parametric model y = θ∗ φ where θ∗ = b0 is the unknown parameter and φ = H(s)u,

H(s) =

s2 + 1 (s + 2)3

According to Theorem 5.2.1, we first need to check the linear independence of H(jω1 ), . . . , H(jωn ). For n = 1 this condition becomes H(jω) 6= 0, ∀ω ∈ R. It is clear that for ω = 1, H(j) = 0, and, therefore, φ may not be PE if we simply choose u to be sufficiently rich of order 1. That is, for u = sin t, the steady-state values of φ, y are equal to zero and, therefore, carry no information about the unknown b0 . We should note, however, that for u = sin ωt and any ω 6= 1, 0, φ is PE. 5

Remark 5.2.1 The above example demonstrates that the condition for the linear independence of the vectors H(jωi ), i = 1, 2, · · · , n on C n is sufficient to guarantee that φ is P E when u is sufficiently rich of order n.

258

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS It also demonstrates that when the plant is partially known, the input u does not have to be sufficiently rich of order n where n is the order of the plant. In this case, the condition on u can be relaxed, depending on the number of the unknown parameters. For further details on the problem of prior information and persistent of excitation, the reader is referred to [35].

In the following sections we use Theorem 5.2.1 to design the input signal u for a wide class of parameter estimators developed in Chapter 4.

5.2.2

Parameter Identifiers with Full-State Measurements

Let us consider the plant x˙ = Ax + Bu, x(0) = x0

(5.2.10)

where C > = I , i.e., the state x ∈ Rn is available for measurement and A, B are constant matrices with unknown elements that we like to identify. We assume that A is stable and u ∈ L∞ . As shown in Chapter 4, the following two types of parameter estimators may be used to estimate A, B from the measurements of x, u. Series-Parallel ˆ x ˆ˙ = Am x ˆ + (Aˆ − Am )x + Bu ˙ ˆ˙ = γ2 ²1 u> Aˆ = γ1 ²1 x> , B

(5.2.11)

where Am is a stable matrix chosen by the designer, ²1 = x − x ˆ , γ 1 , γ2 > 0 are the scalar adaptive gains. Parallel ˆx + Bu ˆ x ˆ˙ = Aˆ ˙ ˆ˙ = γ2 ²1 u> Aˆ = γ1 ²1 x ˆ> , B

(5.2.12)

where ²1 = x − x ˆ and γ1 , γ2 > 0 are the scalar adaptive gains. As shown in Chapter 4, if A is a stable matrix and u ∈ L∞ then ˙ˆ ˆ ˆ ∈ L∞ ; kA(t)k, ˆ˙ x ˆ, A, B kB(t)k, ²1 ∈ L2 ∩ L∞ and ²1 (t) and the elements ˙ˆ ˙ ˆ of A(t), B(t) converge to zero as t → ∞.

5.2. PARAMETER IDENTIFIERS

259

For the estimators (5.2.11) and (5.2.12) to become parameter identifiers, ˆ ˆ the input signal u has to be chosen so that A(t), B(t) converge to the unknown plant parameters A, B, respectively, as t → ∞. For simplicity let us first consider the case where u is a scalar input, i.e., B ∈ Rn×1 . Theorem 5.2.2 Let (A, B) be a controllable pair. If the input u ∈ R1 is sufˆ B ˆ generated by (5.2.11) or ficiently rich of order n + 1, then the estimates A, (5.2.12) converge exponentially fast to the unknown plant parameters A, B, respectively. The proof of Theorem 5.2.2 is quite long and is presented in Section 5.6. An example of a sufficiently rich input u for the estimators (5.2.11), (5.2.12) is the input u=

m X

Ai sin ωi t

i=1

for some constants Ai 6= 0 and ωi 6= ωk for i 6= k and for some integer m ≥ n+1 2 . Example 5.2.2 Consider the second-order plant x˙ = Ax + Bu where x = [x1 , x2 ]> , and the· matrices A, ¸ B are unknown, and A is a stable matrix. −1 0 Using (5.2.11) with Am = , the series-parallel parameter identifier is 0 −1 given by ¸ · ¸ · ¸ · ˆb1 (t) −1 0 a ˆ11 (t) a ˆ12 (t) ˙x u ˆ= (ˆ x − x) + x+ ˆ 0 −1 a ˆ21 (t) a ˆ22 (t) b2 (t) where x ˆ = [ˆ x1 , x ˆ2 ]> and a ˆ˙ ik = (xi − x ˆi )xk , i = 1, 2; k = 1, 2 ˆb˙ i = (xi − x ˆi )u, i = 1, 2 The input u is selected as u = 5 sin 2.5t + 6 sin 6.1t

260

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

which has more frequencies than needed since it is sufficiently rich of order 4. An input with the least number of frequencies that is sufficiently rich for the plant considered is u = c0 + sin ω0 t

5

for some c0 6= 0 and ω0 6= 0.

If u is a vector, i.e., u ∈ Rq , q > 1 then the following theorem may be used to select u. Theorem 5.2.3 Let (A, B) be a controllable pair. If each element ui , i = 1, 2, . . . q of u is sufficiently rich of order n + 1 and uncorrelated, i.e., each ui ˆ ˆ contains different frequencies, then A(t), B(t) converge to A, B, respectively, exponentially fast. The proof of Theorem 5.2.3 is similar to that for Theorem 5.2.2, and is given in Section 5.6. The controllability of the pair (A, B) is critical for the results of Theorems 5.2.2 and 5.2.3 to hold. If (A, B) is not a controllable pair, then the elements of (A, B) that correspond to the uncontrollable part cannot be learned from the output response because the uncontrollable parts decay to zero exponentially fast and are not affected by the input u. The complexity of the parameter identifier may be reduced if some of the elements of the matrices (A, B) are known. In this case, the order of the adaptive law can be reduced to be equal to the number of the unknown parameters. In addition, the input u may not have to be sufficiently rich of order n + 1. The details of the design and analysis of such schemes are left as exercises for the reader and are included in the problem section. In the following section we extend the results of Theorem 5.2.2 to the case where only the output of the plant, rather than the full state, is available for measurement.

5.2.3

Parameter Identifiers with Partial-State Measurements

In this section we concentrate on the SISO plant x˙ = Ax + Bu, >

y = C x

x(0) = x0

(5.2.13)

5.2. PARAMETER IDENTIFIERS

261

where A is a stable matrix, and y, u ∈ R1 are the only signals available for measurement. Equation (5.2.13) may be also written as y = C > (sI − A)−1 Bu + C > (sI − A)−1 x0

(5.2.14)

where, because of the stability of A, ²t = L−1 {C > (sI − A)−1 }x0 is an exponentially decaying to zero term. We would like to design an on-line parameter identifier to estimate the parameters A, B, C. The triple (A, B, C) contains n2 + 2n unknown parameters to be estimated using only input/output data. The I/O properties of the plant (5.2.14) at steady state (where ²t = 0), however, are uniquely determined by at most 2n parameters. These parameters correspond to the coefficients of the transfer function y(s) bm sm + bm−1 sm−1 + . . . + b0 = C > (sI − A)−1 B = u(s) sn + an−1 sn−1 + . . . + a0

(5.2.15)

where m ≤ n − 1. Because there is an infinite number of triples (A, B, C) that give the same transfer function (5.2.15), the triple (A, B, C) associated with the specific or physical state space representation in (5.3.13) cannot be determined, in general, from input-output data. The best we can do in this case is to estimate the n + m + 1 ≤ 2n coefficients of the plant transfer function (5.2.15) and try to calculate the corresponding triple (A, B, C) using some a priori knowledge about the structure of (A, B, C). For example, when (A, B, C) is in a canonical form, (A, B, C) can be uniquely determined from the coefficients of the transfer function by using the results of Chapter 2. In this section, we concentrate on identifying the n + m + 1 coefficients of the transfer function of the plant (5.2.13) rather than the n2 + 2n parameters of the triple (A, B, C). The first step in the design of the parameter identifier is to develop an adaptive law that generates estimates for the unknown plant parameter vector θ∗ = [bm , bm−1 , . . . , b0 , an−1 , an−2 , . . . , , a0 ]> that contains the coefficients of the plant transfer function. As we have shown in Section 2.4.1, the vector θ∗ satisfies the plant parametric equation z = θ∗> φ + ²t

(5.2.16)

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CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

where "

z = ²t =

> (s) α> (s) sn αm y, φ = u, − n−1 y Λ(s) Λ(s) Λ(s)

C > adj(sI − A) x0 , Λ(s)

#>

αi (s) = [si , si−1 , . . . , 1]>

and Λ(s) is an arbitrary monic Hurwitz polynomial of degree n. Using (5.2.16), we can select any one of the adaptive laws presented in Tables 4.2, 4.3, and 4.5 of Chapter 4 to estimate θ∗ . As an example, consider the gradient algorithm θ˙ = Γ²φ

(5.2.17) >

² = z − zˆ,

zˆ = θ φ

where Γ = Γ> > 0 is the adaptive gain matrix, and θ is the estimate of θ∗ at time t. The normalizing signal m2 in this case is chosen as m2 = 1 due to φ ∈ L∞ . The signal vector φ and signal z may be generated by the state equation φ˙ 0 = Λc φ0 + lu,

φ0 (0) = 0

φ1 = P0 φ0 φ˙ 2 = Λc φ2 − ly,

φ2 (0) = 0

φ = z =

(5.2.18)

> > [φ> 1 , φ2 ] y + λ> φ2

where φ0 ∈ Rn , φ1 ∈ Rm+1 , φ2 ∈ Rn 



−λ>

  Λc =  − − − − −  ,

In−1 | 0

   l=  

1 0 .. .

     

0 

h

i

P0 = O(m+1)×(n−m−1) | Im+1 ∈ R(m+1)×n ,

 

λ=  

λn−1 λn−2 .. . λ0

     

5.2. PARAMETER IDENTIFIERS

263

Ii is the identity matrix of dimension i×i, Oi×k is a matrix of dimension i by k with all elements equal to zero, and det(sI − Λc ) = Λ(s) = sn + λ> αn−1 (s). When m = n − 1, the matrix P0 becomes the identity matrix of dimension n × n. The state equations (5.2.18) are developed by using the identity (sI − Λc )−1 l =

αn−1 (s) Λ(s)

established in Chapter 2. The adaptive law (5.2.17) or any other one obtained from Tables 4.2 and 4.3 in Chapter 4 guarantees that ², θ, zˆ ∈ L∞ ; ², θ˙ ∈ L2 ∩ L∞ and ˙ ²(t), θ(t) → 0 as t → ∞ for any piecewise bounded input u. For these adaptive laws to become parameter identifiers, the input signal u has to be chosen to be sufficiently rich so that φ is P E, which in turn guarantees that the estimated parameters converge to the actual ones. Theorem 5.2.4 Assume that the plant transfer function in (5.2.15) has no zero-pole cancellations. If u is sufficiently rich of order n + m + 1, then the adaptive law (5.2.17) or any other adaptive law from Tables 4.2, 4.3, and 4.5 of Chapter 4, based on the plant parametric model (5.2.16), guarantees that the estimated parameter vector θ(t) converges to θ∗ . With the exception of the pure least-squares algorithm in Table 4.3 where the convergence is asymptotic, all the other adaptive laws guarantee exponential convergence of θ to θ∗ . Proof In Chapter 4, we have established that θ(t) converges to θ∗ if the signal φ is PE. Therefore we are left to show that φ is PE if u is sufficiently rich of order n + m + 1 and the transfer function has no zero-pole cancellations. From the definition of φ, we can write φ = H(s)u and H(s)

= =

¤> 1 £ > > αm (s), −αn−1 G(s) Λ(s) £ > ¤> 1 > αm (s)R(s), −αn−1 (s)Z(s) Λ(s)R(s)

(5.2.19)

264

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

Z(s) where G(s) = R(s) is the transfer function of the plant. We first show by contradiction that {H(jω1 ), H(jω2 ), . . . , H(jωn+m+1 )} are linearly independent in C n+m+1 for any ω1 , . . . , ωn+m+1 ∈ R and ωi 6= ωk , i, k = 1, . . . , n + m + 1. Let us assume that there exist ω1 ,. . . ,ωn+m+1 such that H(jω1), . . . ,H(jωn+m+1) are linearly dependent, that is, there exists a vector h = [cm , cm−1 , . . ., c0 , dn−1 , dn−2 , . . . , d0 ]> ∈ C n+m+1 such that   H > (jω1 )   H > (jω2 )   (5.2.20)  h = 0n+m+1  ..   . H > (jωn+m+1 )

where 0n+m+1 is a zero vector of dimension n + m + 1. Using the expression (5.2.19) for H(s), (5.2.20) can be written as 1 [b(jωi )R(jωi ) + a(jωi )Z(jωi )] = 0, i = 1, 2, . . . , n + m + 1 Λ(jωi )R(jωi ) where

b(s) = cm sm + cm−1 sm−1 + · · · + c0 a(s) = −dn−1 sn−1 − dn−2 sn−2 − · · · − d0

Now consider the following polynomial: 4

f (s) = a(s)Z(s) + b(s)R(s) Because f (s) has degree of at most m + n and it vanishes at s = jωi , i = 1, . . . , n + m + 1 points, it must be identically equal to zero, that is a(s)Z(s) + b(s)R(s) = 0 or equivalently G(s) =

Z(s) b(s) =− R(s) a(s)

(5.2.21)

for all s. However, because a(s) has degree at most n − 1, (5.2.21) implies that G(s) has at least one zero-pole cancellation, which contradicts the assumption of no zeropole cancellation in G(s). Thus, we have proved that H(jωi ), i = 1, . . . , n + m + 1 are linearly independent for any ωi 6= ωk , i, k = 1, . . . , n+m+1. It then follows from Theorem 5.2.1 and the assumption of u being sufficiently rich of order n + m + 1 that φ is PE and therefore θ converges to θ∗ . As shown in Chapter 4, the convergence of θ to θ∗ is exponential for all the adaptive laws of Tables 4.2, 4.3, and 4.5 with the exception of the pure least-squares where the convergence is asymptotic. 2

5.2. PARAMETER IDENTIFIERS

265

An interesting question to ask at this stage is what happens to the parameter estimates when u is sufficiently rich of order less than n + m + 1, i.e., when its spectrum Su (ω) is concentrated on k < n + m + 1 points. Let us try to answer this question by considering the plant parametric equation (5.2.16) with ²t = 0, i.e., z = θ∗> φ = θ∗> H(s)u where H(s) is given by (5.2.19). The estimated value of z is given by zˆ = θ> φ = θ> H(s)u The adaptive laws of Tables 4.2, 4.3, and 4.5 of Chapter 4 based on the parametric equation (5.2.16) guarantee that zˆ(t) → z(t) as t → ∞ for any bounded input u. If u contains frequencies at ω1 , ω2 , . . . , ωk , then as t → ∞, zˆ = θ> H(s)u becomes equal to z = θ∗> H(s)u at these frequencies. Therefore, as t → ∞, θ must satisfy the equation Q (jω1 , . . . , jωk ) θ = Q (jω1 , . . . , jωk ) θ∗ where





H > (jω1 )   Q (jω1 , . . . , jωk ) =  ...  > H (jωk )

is a k × n + m + 1 matrix with k linearly independent row vectors. Hence, Q (jω1 , . . . , jωk ) (θ − θ∗ ) = 0 which is satisfied for any θ for which θ − θ∗ is in the null space of Q. If k < n + m + 1 it follows that the null space of Q contains points for which θ 6= θ∗ . This means that zˆ can match z at k frequencies even when θ 6= θ∗ . The following Theorem presented in [201] gives a similar result followed by a more rigorous proof. Theorem 5.2.5 (Partial Convergence) Assume that the plant transfer function in (5.2.15) has no zero-pole cancellations. If u is stationary, then lim Rφ (0) (θ(t) − θ∗ ) = 0

t→∞

where θ(t) is generated by (5.2.17) or any other adaptive law from Tables 4.2, 4.3, and 4.5 of Chapter 4 based on the plant parametric model (5.2.16).

266

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

The proof of Theorem 5.2.5 is given in Section 5.6. Theorem 5.2.5 does not imply that θ(t) converges to a constant let alone ˜ = θ(t) − θ∗ converges to the null to θ∗ . It does imply, however, that θ(t) space of the autocovariance Rφ (0) of φ, which depends on H(s) and the spectrum of u. In fact it can be shown [201] that Rφ (0) is related to the spectrum of u via the equation Rφ (0) =

k X

H(−jωi )H > (jωi )Su (ωi )

i=1

which indicates the dependence of the null space of Rφ (0) on the spectrum of the input u. Example 5.2.3 Consider the second order plant y=

s2

b0 u + a1 s + a0

where a1 , a0 > 0 and b0 6= 0 are the unknown plant parameters. We first express the plant in the form of z = θ∗> φ h i> [s,1] 1 s2 y, φ = Λ(s) u, − Λ(s) y and choose Λ(s) = (s + 2)2 . where θ∗ = [b0 , a1 , a0 ]> , z = Λ(s) Let us choose the pure least-squares algorithm from Table 4.3, i.e., θ˙ P˙ ²

= P ²φ, θ(0) = θ0 = −P φφ> P, P (0) = p0 I = z − θ> φ

> > where θ is the estimate of θ∗ and select p0 = 50. The signal vector φ = [φ> 1 , φ2 ] is generated by the state equations

φ˙ 0 φ1 φ˙ 2 z ·

= Λc φ0 + lu = [0 1]φ0 = Λc φ2 − ly = y + [4 4]φ2 · ¸ ¸ −4 −4 1 where Λc = ,l = . The reader can demonstrate via computer 1 0 0 simulations that: (i) for u = 5 sin t + 12 sin 3t, θ(t) → θ∗ as t → ∞; (ii) for u = 12 sin 3t, θ(t) → θ¯ as t → ∞ where θ¯ is a constant vector that depends on the initial condition θ(0). 5

5.3. ADAPTIVE OBSERVERS

267

Remark 5.2.2 As illustrated with Example 5.2.3, one can choose any one of the adaptive laws presented in Tables 4.2, 4.3, and 4.5 to form parameter identifiers. The complete proof of the stability properties of such parameter identifiers follows directly from the results of Chapter 4 and Theorem 5.2.4. The reader is asked to repeat some of the stability proofs of parameter identifiers with different adaptive laws in the problem section.

5.3

Adaptive Observers

Consider the LTI SISO plant x˙ = Ax + Bu,

x(0) = x0

(5.3.1)

>

y = C x where x ∈ Rn . We assume that u is a piecewise continuous and bounded function of time, and A is a stable matrix. In addition we assume that the plant is completely controllable and completely observable. The problem is to construct a scheme that estimates both the parameters of the plant, i.e., A, B, C as well as the state vector x using only I/O measurements. We refer to such a scheme as the adaptive observer. A good starting point for choosing the structure of the adaptive observer is the state observer, known as the Luenberger observer, used to estimate the state vector x of the plant (5.3.1) when the parameters A, B, C are known.

5.3.1

The Luenberger Observer

If the initial state vector x0 is known, the estimate x ˆ of x in (5.3.1) may be generated by the state observer x ˆ˙ = Aˆ x + Bu, x ˆ(0) = x0

(5.3.2)

where x ˆ ∈ Rn . Equations (5.3.1) and (5.3.2) imply that x ˆ(t) = x(t), ∀t ≥ 0. When x0 is unknown and A is a stable matrix, the following state observer may be used to generate the estimate x ˆ of x: x ˆ˙ = Aˆ x + Bu,

x ˆ(0) = x ˆ0

(5.3.3)

268

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

In this case, the state observation error x ˜=x−x ˆ satisfies the equation x ˜˙ = A˜ x,

x ˜(0) = x0 − x ˆ0

which implies that x ˜(t) = eAt x ˜(0). Because A is a stable matrix x ˜(t) → 0, i.e., x ˆ(t) → x(t) as t → ∞ exponentially fast with a rate that depends on the location of the eigenvalues of A. The observers (5.3.2), (5.3.3) contain no feedback terms and are often referred to as open-loop observers. When x0 is unknown and A is not a stable matrix, or A is stable but the state observation error is required to converge to zero faster than the rate with which k eAt k goes to zero, the following observer, known as the Luenberger observer, is used: x ˆ˙ = Aˆ x + Bu + K(y − yˆ), x ˆ(0) = x ˆ0

(5.3.4)

>

yˆ = C x ˆ where K is a matrix to be chosen by the designer. In contrast to (5.3.2) and (5.3.3), the Luenberger observer (5.3.4) has a feedback term that depends on the output observation error y˜ = y − yˆ. The state observation error x ˜=x−x ˆ for (5.3.4) satisfies x ˜˙ = (A − KC > )˜ x,

x ˜(0) = x0 − x ˆ0

(5.3.5)

Because (C, A) is an observable pair, we can choose K so that A − KC > is a stable matrix. In fact, the eigenvalues of A − KC > , and, therefore, the rate of convergence of x ˜(t) to zero can be arbitrarily chosen by designing K appropriately [95]. Therefore, it follows from (5.3.5) that xˆ(t) → x(t) exponentially fast as t → ∞, with a rate that depends on the matrix A − KC > . This result is valid for any matrix A and any initial condition x0 as long as (C, A) is an observable pair and A, C are known. Example 5.3.1 Consider the plant described by ¸ ¸ · · 1 −4 1 x+ u x˙ = 3 −4 0 y

=

[1, 0]x

The Luenberger observer for estimating the state x is given by · ¸ · ¸ · ¸ −4 1 1 k1 ˙x ˆ = x ˆ+ u+ (y − yˆ) −4 0 3 k2 yˆ =

[1, 0]ˆ x

5.3. ADAPTIVE OBSERVERS where K = [k1 , k2 ]

269

>

is chosen so that · ¸ · ¸ · −4 1 k1 −4 − k1 A0 = − [1 0] = −4 0 k2 −4 − k2

1 0

¸

is a stable matrix. Let us assume that x ˆ(t) is required to converge to x(t) faster than e−5t . This requirement is achieved by choosing k1 , k2 so that the eigenvalues of A0 are real and less than −5, i.e., we choose the desired eigenvalues of A0 to be λ1 = −6, λ2 = −8 and design k1 , k2 so that det(sI − A0 ) = s2 + (4 + k1 )s + 4 + k2 = (s + 6)(s + 8) which gives k1 = 10, k2 = 44

5.3.2

5

The Adaptive Luenberger Observer

Let us now consider the problem where both the state x and parameters A, B, C are to be estimated on-line simultaneously using an adaptive observer. A straightforward procedure for choosing the structure of the adaptive observer is to use the same equation as the Luenberger observer in (5.3.4), ˆ B, ˆ C, ˆ but replace the unknown parameters A, B, C with their estimates A, respectively, generated by some adaptive law. The problem we face with this procedure is the inability to estimate uniquely the n2 + 2n parameters of A, B, C from input/output data. As explained in Section 5.2.3, the best we can do in this case is to estimate the n + m + 1 ≤ 2n parameters of the plant ˆ B, ˆ C. ˆ These calculations, transfer function and use them to calculate A, however, are not always possible because the mapping of the 2n estimated ˆ B, ˆ Cˆ parameters of the transfer function to the n2 + 2n parameters of A, is not unique unless (A, B, C) satisfies certain structural constraints. One such constraint is that (A, B, C) is in the observer form, i.e., the plant is represented as 

.. . In−1   x˙ α =  −ap ... . . .  .. . 0 y = [1 0 . . . 0] xα

    xα + bp u 

(5.3.6)

270

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

where ap = [an−1 , an−2 , . . . a0 ]> and bp = [bn−1 , bn−2 , . . . b0 ]> are vectors of dimension n and In−1 ∈ R(n−1)×(n−1) is the identity matrix. The elements of ap and bp are the coefficients of the denominator and numerator, respectively, of the transfer function y(s) bn−1 sn−1 + bn−2 sn−2 + . . . + b0 = u(s) sn + an−1 sn−1 + . . . a0

(5.3.7)

and can be estimated on-line from input/output data by using the techniques described in Chapter 4. Because both (5.3.1) and (5.3.6) represent the same plant, we can assume the plant representation (5.3.6) and estimate xα instead of x. The disadvantage is that in a practical situation x may represent physical variables that are of interest, whereas xα may be an artificial state variable. The adaptive observer for estimating the state xα of (5.3.6) is motivated from the structure of the Luenberger observer (5.3.4) and is given by ˆ x + ˆbp (t)u + K(t)(y − yˆ) x ˆ˙ = A(t)ˆ

(5.3.8)

yˆ = [1 0 . . . 0]ˆ x where x ˆ is the estimate of xα ,  

ˆ = A(t)  

.. . In−1 .. −ˆ ap (t) . . . . .. . 0

   ˆp (t)  , K(t) = a∗ − a 

a∗ ∈ Rn is chosen so that   

A∗ =  −a∗ 

.. . In−1 .. . ... .. . 0

    

(5.3.9)

is a stable matrix and a ˆp (t) and ˆbp (t) are the estimates of the vectors ap and bp , respectively, at time t. A wide class of adaptive laws may be used to generate a ˆp (t) and ˆbp (t) on-line. As an example, we can start with (5.3.7) to obtain as in Section 2.4.1 the parametric model z = θ∗> φ (5.3.10)

5.3. ADAPTIVE OBSERVERS

271

where "

φ = z =

> (s) αn−1 α> (s) u, − n−1 y Λ(s) Λ(s) n s y = y + λ> φ2 Λ(s)

#>

h

> = φ> 1 , φ2

i>

Λ(s) = sn + λ> αn−1 (s) and θ∗ = [bn−1 , bn−2 , . . . , an−1 , an−2 , . . . , a0 ]> is the parameter vector to be estimated and Λ(s) is a Hurwitz polynomial of degree n chosen by the designer. A state-space representation for φ and z 4 (s) may be obtained as in (5.2.18) by using the identity (sI − Λc )−1 l = αn−1 Λ(s) where (Λc , l) is in the controller canonical form and det(sI − Λc ) = Λ(s). In view of (5.3.10), we can choose any adaptive law from Tables 4.2, 4.3 and 4.5 of Chapter 4 to estimate θ∗ and, therefore, ap , bp on-line. We can form a wide class of adaptive observers by combining (5.3.8) with any adaptive law from Tables 4.2, 4.3 and 4.5 of Chapter 4 that is based on the parametric plant model (5.3.10). We illustrate the design of such adaptive observer by using the gradient algorithm of Table 4.2 (A) in Chapter 4 as the adaptive law. The main equations of the observer are summarized in Table 5.1. The stability properties of the class of adaptive observers formed by combining the observer equation (5.3.8) with an adaptive law from Tables 4.2 and 4.3 of Chapter 4 are given by the following theorem. Theorem 5.3.1 An adaptive observer for the plant (5.3.6) formed by combining the observer equation (5.3.8) and any adaptive law based on the plant parametric model (5.3.10) obtained from Tables 4.2 and 4.3 of Chapter 4 guarantees that (i) All signals are u.b. (ii) The output observation error y˜ = y − yˆ converges to zero as t → ∞. (iii) If u is sufficiently rich of order 2n, then the state observation error 4 x ˜ = xα − x ˆ and parameter error θ˜ = θ − θ∗ converge to zero. The rate of convergence is exponential for all adaptive laws except for the pure least-squares where the convergence is asymptotic.

272

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS Table 5.1 Adaptive observer with gradient algorithm 

Plant

.. . In−1   x˙ α =  −ap ... . . .  .. . 0 y = [1, 0 . . . 0]xα

    xα + bp u, xα ∈ Rn 

 .. .I n−1     ˆ .. x ˆ˙ =−ˆ ˆ+ bp (t)u+(a∗ −ˆ ap (t))(y− yˆ) x  ap (t) . . . .  .. . 0 yˆ = [1 0 . . . 0]ˆ x 

Observer

Adaptive law

θ˙ = Γ²φ h i> > (t) θ = ˆb> (t), a ˆ , ²= p p > > zˆ = θ φ, Γ = Γ > 0 ·

φ= z=

Design variables

α> α> n−1 (s) n−1 (s) Λ(s) u, − Λ(s) y sn Λ(s) y

z−ˆ z , m2

¸>

a∗ is chosen so that A∗ in (5.3.9) is stable; m2 = 1 or m2 = 1 + φ> φ; Λ(s) is a monic Hurwitz polynomial of degree n

Proof (i) The adaptive laws of Tables 4.2 and 4.3 of Chapter 4 guarantee that ², ²m, θ˙ ∈ L2 ∩L∞ and θ ∈ L∞ independent of the boundedness of u, y. Because u ∈ L∞ and the plant is stable, we have xα , y, φ, m ∈ L∞ . Because of the boundedness of y, φ we can also establish that ², ²m, θ˙ → 0 as t → ∞ by showing that ²˙ ∈ L∞ ˙ φ˙ ∈ L∞ ), which, together with ² ∈ L2 , implies that ² → 0 as (which follows from θ, t → ∞ (see Lemma 3.2.5). Because m, φ ∈ L∞ , the convergence of ²m, θ˙ to zero follows. The proof of (i) is complete if we establish that x ˆ ∈ L∞ . We rewrite the observer

5.3. ADAPTIVE OBSERVERS

273

equation (5.3.8) in the form ˆ − A∗ )xα x ˆ˙ = A∗ x ˆ + ˆbp (t)u + (A(t) (5.3.11) h i> Because θ = ˆb> ˆ> , u, xα ∈ L∞ and A∗ is a stable matrix, it follows that p (t), a p (t) x ˆ ∈ L∞ . Hence, the proof of (i) is complete. 4

(ii) Let x ˜ = xα − x ˆ be the state observation error. It follows from (5.3.11), (5.3.6) that x ˜˙ = A∗ x ˜ − ˜bp u + a ˜p y, x ˜(0) = xα (0) − x ˆ(0) (5.3.12) 4

4

where ˜bp = ˆbp − bp , a ˜p = a ˆp − ap . From (5.3.12), we obtain y˜ = C > x ˜(s) = C > (sI − A∗ )−1 (−˜bp u + a ˜p y) + ²t © ª where ²t = L−1 C > (sI − A∗ )−1 x ˜(0) is an exponentially decaying to zero term. Because (C, A) is in the observer canonical form, we have C > (sI − A∗ )−1 =

> αn−1 (s) [sn−1 , sn−2 , . . . s, 1] = ∗ det(sI − A ) det(sI − A∗ )

4

Letting Λ∗ (s) = det(sI − A∗ ), we have i 1 X n−i h ˜ s − b u + a ˜ y + ²t n−i n−i Λ∗ (s) i=1 n

y˜(s) =

where ˜bi , a ˜i is the ith element of ˜bp and a ˜p , respectively, which may be written as i Λ(s) X sn−i h ˜ y˜(s) = ∗ −bn−i u + a ˜n−i y + ²t Λ (s) i=1 Λ(s) n

(5.3.13)

where Λ(s) is the Hurwitz polynomial of degree n defined in (5.3.10). We now apply Lemma A.1 (see Appendix A) for each term under the summation in (5.3.13) to obtain n · Λ(s) X ˜ sn−i sn−i y˜ = − b u + a ˜ y n−i n−i Λ∗ (s) i=1 Λ(s) Λ(s) i ˙ −Wci (s) (Wbi (s)u) ˜bn−i + Wci (s) (Wbi (s)y) a ˜˙ n−i + ²t (5.3.14) where the elements of Wci (s), Wbi (s) are strictly proper transfer functions with the same poles as Λ(s). Using the definition of φ and parameter error θ˜ = θ − θ∗ , we rewrite (5.3.14) as ( ) n ³ ´ X Λ(s) ˙ > ˜ ˜ ˙ y˜ = ∗ −θ φ + Wci (s) − (Wbi (s)u) bn−i + (Wbi (s)y) a ˜n−i + ²t Λ (s) i=1

274

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

u

-

Plant

- Parameter ¾

y

-

Luenberger Observer (Equation 5.3.8)

x ˆ-

6

Estimation

(ˆ ap , ˆbp ) Figure 5.1 General structure of the adaptive Luenberger observer. ˙ Because ˜bn−i , a ˜˙ n−i ∈ L2 ∩ L∞ converge to zero as t → ∞, u, y ∈ L∞ and the elements of Wci (s), Wbi (s) are strictly proper stable transfer functions, it follows from Corollary 3.3.1 that all the terms under the summation are in L2 ∩ L∞ and converge to zero as t → ∞. Furthermore, from ²m2 = z − zˆ = −θ˜> φ, m ∈ L∞ , ² ∈ L2 ∩ L∞ and ²(t) → 0 as t → 0, we have that θ˜> φ ∈ L2 ∩ L∞ converges to zero as t → ∞. Hence, y˜ is the sum of an output of a proper stable transfer function whose input is in L2 and converges to zero as t → ∞ and the exponentially decaying to zero term ²t . Therefore, y˜(t) → 0 as t → ∞. (iii) If φ is P E, then we can establish, using the results of Chapter 4, that a ˜p (t), ˜bp (t) converge to zero. Hence, the input −˜bp u + a ˜p y converges to zero, which, together with the stability of A∗ , implies that x ˜(t) → 0. With the exception of the pure least-squares, all the other adaptive laws guarantee that the convergence of ˜bp , a ˜p to zero is exponential, which implies that x ˜ also goes to zero exponentially fast . The PE property of φ is established by using exactly the same steps as in the proof of Theorem 5.2.4. 2

The general structure of the adaptive observer is shown in Figure 5.1. The only a priori knowledge assumed about the plant (5.3.1) is that it is completely observable and completely controllable and its order n is known. The knowledge of n is used to choose the order of the observer, whereas the observability of (C, A) is used to guarantee the existence of the state space representation of the plant in the observer form that in turn enables us to design a stable adaptive observer. The controllability of (A, B) is not needed for stability, but it is used together with the observability of (C, A) to establish that φ is P E from the properties of the input u.

5.3. ADAPTIVE OBSERVERS

275

Theorem 5.3.1 shows that for the state xα of the plant to be estimated exactly, the input has to be sufficiently rich of order 2n, which implies that the adaptive law has to be a parameter identifier. Even with the knowledge of the parameters ap , bp and of the state xα , however, it is not in general possible to calculate the original state of the plant x because of the usual nonunique mapping from the coefficients of the transfer function to the parameters of the state space representation. Example 5.3.2 Let us consider the second order plant · ¸ · ¸ −a1 1 b1 x˙ = x+ u −a0 0 b0 y

= [1, 0] x

where a1 , a0 , b1 , b0 are the unknown parameters and u, y are the only signals available for measurement. Using Table 5.1, the adaptive observer for estimating x, and the unknown parameters are described as follows: The observer equation is given by ¸ · ¸ · ¸ · ˆb1 (t) 9−a ˆ1 (t) −ˆ a1 (t) 1 ˙x u+ (y − yˆ) ˆ = x ˆ+ ˆ 20 − a ˆ0 (t) −ˆ a0 (t) 0 b0 (t) yˆ =

[1, 0] x ˆ ·

where the constants a∗1 = 9, a∗0 = 20 are selected so that A∗ = eigenvalues at λ1 = −5, λ2 = −4. The adaptive law is designed by first selecting Λ(s) = (s + 2)(s + 3) = s2 + 5s + 6 and generating the information vector · −5 φ˙ 1 = 1 · −5 φ˙ 2 = 1

> > φ = [φ> 1 , φ2 ] ¸ · ¸ −6 1 φ1 + u 0 0 ¸ · ¸ −6 −1 φ2 + y 0 0

and the signals z

=

zˆ =

y + [5, 6]φ2 h i ˆb1 , ˆb0 φ1 + [ˆ a1 , a ˆ0 ] φ2

² =

z − zˆ

−a∗1 −a∗0

1 0

¸ has

276

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

The adaptive law is then given by # " ˆb˙ 1 = γ1 ²φ1 , ˆb˙ 2

·

a ˆ˙ 1 a ˆ˙ 2

¸ = γ2 ²φ2

The adaptive gains γ1 , γ2 > 0 are usually chosen by trial and error using simulations in order to achieve a good rate of convergence. Small γ1 , γ2 may result in slow convergent rate whereas large γ1 , γ2 may make the differential equations “stiff” and difficult to solve numerically on a digital computer. In order for the parameters to converge to their true values, the plant input u is chosen to be sufficiently rich of order 4. One possible choice for u is u = A1 sin ω1 t + A2 sin ω2 t for some constants A1 , A2 6= 0 and ω1 6= ω2 .

5.3.3

5

Hybrid Adaptive Luenberger Observer

The adaptive law for the adaptive observer presented in Table 5.1 can be replaced with a hybrid one without changing the stability properties of the observer in any significant way. The hybrid adaptive law updates the parameter estimates only at specific instants of time tk , where tk is an unbounded monotonic sequence, and tk+1 − tk = Ts where Ts may be considered as the sampling period. The hybrid adaptive observer is developed by replacing the continuoustime adaptive law with the hybrid one given in Table 4.5 as shown in Table 5.2. The combination of the discrete-time adaptive law with the continuoustime observer equation makes the overall system more difficult to analyze. The stability properties of the hybrid adaptive observer, however, are very similar to those of the continuous-time adaptive observer and are given by the following theorem. Theorem 5.3.2 The hybrid adaptive Luenberger observer presented in Table 5.2 guarantees that (i)

All signals are u.b.

(ii)

The output observation error y˜ = y − yˆ converges to zero as t → ∞.

4

4

(iii) If u is sufficiently rich of order 2n, then the state observation error x ˜= 4 ∗ xα − x ˆ and parameter error θ˜k = θk − θ converge to zero exponentially fast.

5.3. ADAPTIVE OBSERVERS

277

Table 5.2 Hybrid adaptive Luenberger observer 

.. . In−1   . x˙ α =  −ap .. . . .  .. . 0 y = [1, 0 . . . 0]u

Plant

    xα + bp u, xα ∈ Rn 



 .. . I n−1    ˆ .. x ˆ˙ =−ˆ ˆ+ bk u + (a∗−ˆ ak )(y− yˆ) x  ak . . . .  .. . 0 yˆ = [1, 0, . . . , 0]ˆ x x ˆ ∈ Rn , t ∈ [tk , tk+1 ), k = 0, 1, . . .

Observer

Rt

Hybrid adaptive law

Design variable

θk+1 = θk + Γ tkk+1 ²(τ )φ(τ )dτ z , zˆ(t) = θk> φ(t), ∀t ∈ [tk , tk+1 ) ² = z−ˆ m2 2 m = 1 + αφ> φ, α ≥ 0 ·

φ=

¸>

α> α> n−1 (s) n−1 (s) Λ(s) u,− Λ(s) y

h

sn , z = Λ(s) y, θk = ˆb> ˆ> k ,a k

i>

a∗ is chosen so that A∗ in (5.3.9) is stable; Λ(s) is monic Hurwitz of degree n; Ts = tk+1 − tk is the sampling period; Γ = Γ> > 0, Ts are chosen so that 2 − Ts λmax (Γ) > γ0 for some γ0 > 0

To prove Theorem 5.3.2 we need the following lemma: Lemma 5.3.1 Consider any piecewise constant function defined as f (t) = fk

∀t ∈ [kTs , (k + 1)Ts ), 4

k = 0, 1, . . . .

If the sequence {∆fk } ∈ `2 where ∆fk = fk+1 − fk , then there exists a continuous function f¯(t) such that |f − f¯| ∈ L2 and f˙¯ ∈ L2 .

278

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

Proof The proof of Lemma 5.3.1 is constructive and rather simple. Consider the following linear interpolation: fk+1 − fk f¯(t) = fk + (t − kTs ) Ts

∀t ∈ [kTs , (k + 1)Ts )

It is obvious that f¯ has the following properties: (i) it is continuous; (ii) |f¯(t) − f (t)| ≤ |fk+1 − fk |; (iii) f˙¯ is piecewise continuous and |f˙¯(t)| = T1s |fk+1 − fk | ∀t ∈ [kTs , (k + 1)Ts ). Therefore, using the assumption ∆fk ∈ `2 we have Z

∞

|f¯(t) − f (t)|2 dt ≤ Ts

0

and

∞ X

|∆fk |2 < ∞

k=0

Z

∞

|f˙¯(t)|2 dt ≤

0

∞ X

|∆fk |2 < ∞

k=0

i.e., f˙¯ ∈ L2 .

2

Proof of Theorem 5.3.2 (i) We have shown in Chapter 4 that T the hybrid adaptive law given in Table 4.5 guarantees that θk ∈ l∞ ; ², ²m ∈ L∞ L2 . Because u ∈ L∞ and the plant is stable, we have y, xα , φ, m ∈ L∞ . As we show in the proof of Theorem 5.3.1, we can write the observer equation in the form x ˆ˙ = A∗ x ˆ + ˆbk u + (Aˆk − A∗ )xα , ∀t ∈ [tk , tk+1 )

(5.3.15)

where A∗ is a stable matrix and ¯  ¯ In−1 ¯ 4 Aˆk = −ˆ ak ¯¯ · · ·  ¯ 0 

Because a ˆk , ˆbk ∈ l∞ ; xα , u ∈ L∞ , it follows from (5.3.15) that x ˆ ∈ L∞ , and, therefore, all signals are u.b. (ii) Following the same procedure as in the proof of Theorem 5.3.1, we can express y˜ = y − yˆ as y˜ = C > (sI − A∗ )−1 (−˜bk u + a ˜k y) + ²t

(5.3.16)

where ²t is an exponentially decaying to zero term. From Lemma 5.3.1 and the properties of the hybrid adaptive law, i.e., θ˜k ∈ l∞ and |θ˜k+1 − θ˜k | ∈ `2 , we conclude that there exists a continuous piecewise vector

5.4.

ADAPTIVE OBSERVER WITH AUXILIARY INPUT

279

T ˙ ˜ function θ˜ such that |θ(t)− θ˜k (t)|, θ˜ ∈ L∞ L2 , where θ˜k (t) is the piecewise constant function defined by θ˜k (t) = θ˜k , ∀t ∈ [tk , tk+1 ). Therefore, we can write (5.3.16) as y˜ = C > (sI − A∗ )−1 (−˜bu + a ˜y) + f (t) + ²t where

(5.3.17)

³ ´ f (t) = C > (sI − A∗ )−1 (˜b − ˜bk )u − (˜ a−a ˜k )y

T Using Corollary 3.3.1, it follows from u, y ∈ L∞ and |θ˜ − θ˜k | ∈ L∞ L2 that T f ∈ L∞ L2 and f (t) → 0 as t → ∞. Because now θ˜ has the same properties as those used in the continuous adaptive Luenberger observer, we can follow exactly the same procedure as in the proof of Theorem 5.3.1 to shown that the first term in (5.3.17) converges to zero and, therefore, y˜(t) → 0 as t → ∞. (iii) We have established in the proof of Theorem 5.2.4 that when u is sufficiently rich of order 2n, φ is PE. Using the PE property of φ and Theorem 4.6.1 (iii) we have that θk → θ∗ exponentially fast. 2

5.4

Adaptive Observer with Auxiliary Input

Another class of adaptive observers that attracted considerable interest [28, 123, 172] involves the use of auxiliary signals in the observer equation. An observer that belongs to this class is described by the equation ˆ x + ˆbp (t)u + K(t) (y − yˆ) + v, x ˆ˙ = A(t)ˆ

x ˆ(0) = x ˆ0

yˆ = [1 0 . . . 0]ˆ x

(5.4.1)

ˆbp (t), K(t) are as defined in (5.3.8), and v is an auxiliary vector ˆ where x ˆ, A(t), input to be designed. The motivation for introducing v is to be able to use the SPR-Lyapunov design approach to generate the adaptive laws for ˆbp (t), K(t). This approach is different from the one taken in Section ˆ A(t), 5.3 where the adaptive law is developed independently without considering the observer structure. The first step in the SPR-Lyapunov design approach is to obtain an error equation that relates the estimation or observation error with the parameter error as follows: By using the same steps as in the proof of Theorem 5.3.1, the state error x ˜ = xα − x ˆ satisfies x ˜˙ = A∗ x ˜+a ˜p y − ˜bp u − v,

x ˜(0) = x ˜0

280

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS y˜ = C > x ˜

(5.4.2)

where a ˜p , ˜bp are the parameter errors. Equation (5.4.2) is not in the familiar form studied in Chapter 4 that allows us to choose an appropriate Lyapunov-like function for designing an adaptive law and proving stability. The purpose of the signal vector v is to convert (5.4.2) into a form that is suitable for applying the Lyapunov approach studied in Chapter 4. The following Lemma establishes the existence of a vector v that converts the error equation (5.4.2) into one that is suitable for applying the SPR-Lyapunov design approach. Lemma 5.4.1 There exists a signal vector v ∈ Rn , generated from measurements of known signals, for which the system (5.4.2) becomes e˙ = A∗ e + Bc (−θ˜> φ),

e(0) = x ˜0

>

y˜ = C e

(5.4.3)

where C > (sI − A∗ )−1 Bc is SPR, φ ∈ R2n is a signal vector generated from > n input/output data, θ˜ = [˜b> ˜> p ,a p ] , and e ∈ R is a new state vector. Proof Because (C, A∗ ) is in the observer form, we have C > (sI − A∗ )−1 =

> (s) αn−1 ∗ Λ (s)

where αn−1 (s) = [sn−1 , sn−2 , . . . , 1]> , Λ∗ (s) = det(sI − A∗ ). Therefore, (5.4.2) can be expressed in the form h i y˜ = C > (sI − A∗ )−1 a ˜p y − ˜bp u − v + ²t ( n ) ´ α> (s) Λ(s) X sn−i ³ n−1 (5.4.4) = a ˜n−i y − ˜bn−i u − v + ²t Λ∗ (s) i=1 Λ(s) Λ(s) © ª where ²t = L−1 C > (sI − A∗ )−1 x ˜0 , Λ(s) = sn−1 + λ> αn−2 (s) is a Hurwitz polynomial, λ> = [λn−2 , λn−3 , . . . , λ1 , λ0 ] is to be specified and a ˜i , ˜bi is the ith element ˜ of a ˜p , bp respectively. Applying the Swapping Lemma A.1 given in Appendix A to each term under the summation in (5.4.4), we obtain ( n µ ¶ sn−i sn−i Λ(s) X ˜ a ˜ y˜ = y − b u (5.4.5) n−i n−i Λ∗ (s) i=1 Λ(s) Λ(s) ) n n o α> (s) X ˙ n−1 ˜ ˙ v + ²t + Wci (s)(Wbi (s)y)a ˜n−i − Wci (s)(Wbi (s)u)bn−i − Λ(s) i=1

5.4. ADAPTIVE OBSERVER WITH AUXILIARY INPUT

281

We now use Lemma A.1 to obtain expressions for Wci , Wbi in terms of the parameters of Λ(s). Because Wc1 , Wb1 are the transfer functions resulting from swapping with the transfer function α> (s)λ sn−1 = 1 − n−2 = 1 − C0> (sI − Λ0 )−1 λ = 1 + C0> (sI − Λ0 )−1 (−λ) Λ(s) Λ(s) where C0 = [1, 0, . . . , 0]> ∈ Rn−1    Λ0 =  −λ 

.. . In−2 .. . ... .. . 0

    ∈ R(n−1)×(n−1) 

it follows from Lemma A.1 that Wc1 (s) = −C0> (sI − Λ0 )−1 =

> −αn−2 (s) , Λ(s)

Wb1 (s) = (sI − Λ0 )−1 (−λ)

Similarly Wci , Wbi i = 2, 3, . . . , n result from swapping with sn−i = C0> (sI − Λ0 )−1 di Λ(s) where di = [0, . . . , 0, 1, 0, . . . , 0]> ∈ Rn−1 has all its elements equal to zero except for the (i−1)−th element that is equal to one. Therefore it follows from Lemma A.1 that α> (s) Wci (s) = −C0> (sI − Λ0 )−1 = − n−2 , i = 2, 3, . . . , n Λ(s) Wbi (s) = (sI − Λ0 )−1 di , If we define

·

i = 2, 3, . . . , n

> (s) αn−1 α> (s) φ= u, − n−1 y Λ(s) Λ(s) 4

¸>

and use the expressions for Wci (s), Wbi (s), (5.4.5) becomes ½ Λ(s) sn−1 y˜ = −θ˜> φ − v1 ∗ Λ (s) Λ(s) " n #) o > ¤ £ ¤˙ αn−2 (s) X n£ −1 −1 ˜ ˙ (sI − Λ0 ) di y a ˜n−i − (sI − Λ0 ) di u bn−i + v¯ − + ²t Λ(s) i=1 where v is partitioned as v = [v1 , v¯> ]> with v1 ∈ R1 and v¯ ∈ Rn−2 and d1 = −λ.

282

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS Choosing v1 = 0 and v¯ = −

n n o X £ ¤ £ ¤˙ (sI − Λ0 )−1 di y a ˜˙ n−i − (sI − Λ0 )−1 di u ˜bn−i i=1

we obtain y˜ =

Λ(s) (−θ˜> φ) + ²t Λ∗ (s)

(5.4.6)

Because > αn−1 (s)bλ Λ(s) sn−1 + λ> αn−2 (s) = = = C > (sI − A∗ )−1 bλ ∗ ∗ ∗ Λ (s) Λ (s) Λ (s)

where bλ = [1, λ> ]> ∈ Rn and C = [1, 0, . . . , 0]> ∈ Rn . For Bc = bλ , (5.4.3) is a minimal state-space representation of (5.4.6). Because Λ(s) is arbitrary, its coefficient vector λ can be chosen so that ΛΛ(s) ¯ is implementable ∗ (s) is SPR. The signal v ˙ because a ˜˙ i , ˜bi and u, y are available for measurement. 2

We can nowh use (5.4.3) instead of (5.4.2) to develop an adaptive law for i > > > ˆ . Using the results of Chapter 4, it follows that the generating θ = bp , a ˆp adaptive law is given by θ˙ = Γ²φ,

² = y˜ = y − yˆ

(5.4.7)

where Γ = Γ> > 0. We summarize the main equations of the adaptive observer developed above in Table 5.3. The structure of the adaptive observer is shown in Figure 5.2. Theorem 5.4.1 The adaptive observer presented in Table 5.3 guarantees that for any bounded input signal u, (i) (ii)

all signals are u.b. y˜(t) = y − yˆ → 0 as t → ∞. ˙ (iii) a ˆ˙ p (t), ˆbp (t) ∈ L2 ∩ L∞ and converge to zero as t → ∞. In addition, if u is sufficiently rich of order 2n, then (iv) x ˜(t) = xα (t) − x ˆ(t), a ˜p (t) = a ˆp (t) − ap , ˜bp (t) = ˆbp (t) − bp converge to zero exponentially fast.

5.4. ADAPTIVE OBSERVER WITH AUXILIARY INPUT

283

Table 5.3 Adaptive observer with auxiliary input ¯  ¯ I ¯ n−1  ¯  x˙ α = −ap ¯ · · ·  xα + bp u, ¯ ¯ 0 

Plant

y = [1 0 . . . 0]xα

ap = [an−1 , . . . , a0 ]> , bp = [bn−1 , . . . , b0 ]> ¯  ¯ I ¯ n−1  ¯  ˆ+ˆbp (t)u+(a∗−ˆ ap (t))(y−ˆ y )+v x ˆ˙ =−ˆ ap (t) ¯ · · ·  x ¯ ¯ 0 

Observer

yˆ = [1 0 . . . 0]ˆ x "

Adaptive law

ˆb˙ p a ˆ˙ p

φ=

#

= Γφ(y − yˆ), ·

Γ = Γ> > 0

α> α> n−1 (s) n−1 (s) u, − Λ(s) Λ(s) y

" #

¸>

·

P 0 ˙ , v = ni=1 − [Wi (s)y] a ˆ˙ n−i +[Wi (s)u] ˆbn−i v= v Wi (s) = (sI − Λ0 )−1 di , > d = −λ; di = [0...0, 1, 0...0], i = 2, . . . , n

Auxiliary input

1

¸

↑

(i − 1)

¯  ¯ I ¯ n−2  ¯  Λ0 = −λ ¯ · · ·  , λ = [λn−2 , . . . , λ0 ]> ¯ ¯ 0 

det(sI − Λ0 ) = Λ(s) = sn−1 + λ> αn−2 (s) ¯  ¯I ¯ n−1  ¯  (i) a∗ is chosen such that A∗=−a∗ ¯ · · ·  is stable ¯ ¯ 0 

Design variables

(ii) The vector λ is chosen such " #that Λ0 is stable 1 and [1 0 . . . 0] (sI − A∗ )−1 is SPR. λ

284

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS u

- Plant

ν

y

+? y˜ Σn −6

¾ Auxiliary Signal Generator ¾ Observer x ˆ ¡¢ - (Equation (5.4.1))

C>

yˆ

6

y˜

-

Parameter Estimator

a ˆp , ˆbp

Figure 5.2 Structure of the adaptive observer with auxiliary input signal. Proof The main equations that describe the stability properties of the adaptive observer are the error equation (5.4.3) that relates the parameter error with the state observation error and the adaptive law (5.4.7). Equations (5.4.3) and (5.4.7) are analyzed in Chapter 4 where it has been shown that e, y˜ ∈ L2 ∩ L∞ and θ˜ ∈ L∞ for any signal vector φ with piecewise continuous elements. Because u, y ∈ L∞ , it ˙ e, follows that φ ∈TL∞ and from (5.4.3) and (5.4.7) that θ, ˙ y˜˙ ∈ L∞ and θ˙ ∈ L2 . Using e, y˜ ∈ L2 L∞ together with e, ˙ y˜˙ ∈ L∞ , we have y˜(t) → 0, e(t) → 0 as ˙ → 0 as t → ∞. From θ˙ ∈ L2 ∩ L∞ , it follows that t → ∞, which implies that θ(t) v ∈ L2 ∩ L∞ and v(t) → 0 as t → ∞. Hence, all inputs to the state equation (5.4.2) are in L∞ , which implies that x ˜ ∈ L∞ , i.e., x ˆ ∈ L∞ . In Chapter 4 we established that if φ, φ˙ ∈ L∞ and φ is P E, then the error ˜ → 0 as t → ∞ exponentially fast. equations (5.4.3) and (5.4.7) guarantee that θ(t) ˜ → 0 as In our case φ, φ˙ ∈ L∞ and therefore if u is chosen so that φ is P E then θ(t) ˙ t → ∞ exponentially fast, which implies that e(t), y˜(t), θ(t), v(t), a ˜p (t)y, ˜bp (t)u and, therefore, x ˜(t) converge to zero exponentially fast. To explore the PE property of φ, we note that φ is related to u through the equation · ¸ 1 αn−1 (s) φ= u Λ(s) −αn−1 (s)C > (sI − A)−1 bp Because u is sufficiently rich of order 2n, the P E property of φ can be established by following exactly the same steps as in the proof of Theorem 5.2.4. 2

The auxiliary signal vector v¯ can be generated directly from the signals

5.4. ADAPTIVE OBSERVER WITH AUXILIARY INPUT

285

˜˙ i.e., the filters Wi for y, u do not have to be implemented. This φ and θ, simplification reduces the number of integrators required to generate φ, v¯ considerably and it follows from the relationship (sI − Λ0 )−1 di = Qi

αn−1 (s) , i = 1, 2, . . . , n Λ(s)

where Qi ∈ R(n−1)×(n−1) are constant matrices whose elements depend on the coefficients of the numerator polynomials of (sI − Λ0 )−1 di (see Problem 2.12). As with the adaptive Luenberger observer, the adaptive observer with auxiliary input shown in Table 5.3 requires the input u to be sufficiently rich of order 2n in order to guarantee exact plant state observation. The only difference between the two observers is that the adaptive Luenberger observer may employ any one of the adaptive laws given in Tables 4.2, 4.3, and 4.5 whereas the one with the auxiliary input given in Table 5.3 relies on the SPR-Lyapunov design approach only. It can be shown, however, (see Problem 5.16) by modifying the proof of Lemma 5.4.1 that the observation error y˜ may be expressed in the form y˜ = −θ˜> φ

(5.4.8)

by properly selecting the auxiliary input v and φ. Equation (5.4.8) is in the form of the error equation that appears in the case of the linear parametric model y = θ∗> φ and allows the use of any one of the adaptive laws of Table 4.2, 4.3, and 4.5 leading to a wide class of adaptive observers with auxiliary input. The following example illustrates the design of an adaptive observer with ˙ auxiliary input v and the generation of v from the signals φ, θ. Example 5.4.1 Let us consider the second order plant · ¸ · ¸ −a1 1 b1 x˙ α = xα + u −a0 0 b0 y = [1 0] xα where a1 > 0, a0 > 0, b1 , b0 are unknown constants and y, u are the only available signals for measurement.

286

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

We use Table 5.3 to develop the adaptive observer for estimating xα and the unknown plant parameters. We start with the design variables. We choose a∗ = [4, 4]> , λ = λ0 , Λ(s) = s + λ0 . Setting λ0 = 3 we have · ¸ s+3 1 ∗ −1 = [1 0][sI − A ] 3 (s + 2)2 which is SPR. h i> s 1 s 1 The signal vector φ = s+3 u, s+3 u, − s+3 y, − s+3 y is realized as follows φ = [φ1 , φ2 , φ3 , φ4 ]

>

where φ1 φ2

= =

φ3 φ4

= =

u − 3φ¯1 , φ¯˙ 1 = −3φ¯1 + u φ¯1 −y + 3φ¯3 , φ¯˙ 3 = −3φ¯3 + y −φ¯3

with φ¯1 (0) = 0, φ¯3 (0) = 0. For simplicity we choose the adaptive gain Γ = diag{10, 10, 10, 10}. The adaptive law is given by ˆb˙ 1 = 10φ1 (y − yˆ),

and the signal v¯ = = =

ˆb˙ 0 = 10φ2 (y − yˆ)

a ˆ˙ 1 = 10φ3 (y − yˆ), a ˆ˙ 0 = 10φ4 (y − yˆ) ¸ · 0 by vector v = v¯ µ µ ¶ µ ¶ ¶ µ ¶ 3 1 3 1 ˙ ˙ y a ˆ˙ 1 − u ˆb1 − y a ˆ˙ 0 + u ˆb0 s+3 s+3 s+3 s+3 ˙ ˙ 3a ˆ˙ 1 φ¯3 − 3ˆb1 φ¯1 − a ˆ˙ 0 φ¯3 + ˆb0 φ¯1 10(y − yˆ)(−3φ3 φ4 − 3φ1 φ2 + φ24 + φ22 )

The observer equation becomes · ¸ · ¸ · ¸ ˆb1 −ˆ a1 1 4−a ˆ1 ˙x ˆ = x ˆ+ ˆ u+ (y − yˆ) −ˆ a0 0 4−a ˆ0 b0 · ¸ £ ¤ 0 (y − yˆ) −3φ3 φ4 − 3φ1 φ2 + φ24 + φ22 + 10 The input signal is chosen as u = A1 sin ω1 t + A2 sin ω2 t for some A1 , A2 6= 0 and ω1 6= ω2 . 5

5.5. ADAPTIVE OBSERVERS FOR NONMINIMAL PLANT MODELS287

5.5

Adaptive Observers for Nonminimal Plant Models

The adaptive observers presented in Sections 5.3 and 5.4 are suitable for estimating the states of a minimal state space realization of the plant that is expressed in the observer form. Simpler (in terms of the number of integrators required for implementation) adaptive observers may be constructed if the objective is to estimate the states of certain nonminimal state-space representations of the plant. Several such adaptive observers have been presented in the literature over the years [103, 108, 120, 123, 130, 172], in this section we present only those that are based on the two nonminimal plant representations developed in Chapter 2 and shown in Figures 2.2 and 2.3.

5.5.1

Adaptive Observer Based on Realization 1

Following the plant parameterization shown in Figure 2.2, the plant (5.3.1) is represented in the state space form φ˙ 1 = Λc φ1 + lu, φ˙ 2 = Λc φ2 − ly, ω˙ = Λc ω, η0 =

φ1 (0) = 0 φ2 (0) = 0

ω(0) = ω0 = B0 x0

(5.5.1)

C0> ω

z = y + λ> φ2 = θ∗> φ + η0 y = θ∗> φ − λ> φ2 + η0 > > n n×n is a known stable where ω ∈ Rn , φ = [φ> 1 , φ2 ] , φi ∈ R , i = 1, 2; Λc ∈ R matrix in the controller form; l = [1, 0, . . . , 0]> ∈ Rn is a known vector such (s) that (sI − Λc )−1 l = αn−1 and Λ(s) = det(sI − Λc ) = sn + λ> αn−1 (s), Λ(s) λ = [λn−1 , . . . , λ0 ]> ; θ∗ = [bn−1 , bn−2 , . . . , b0 , an−1 , an−2 , . . . , a0 ]> ∈ R2n are the unknown parameters to be estimated; and B0 ∈ Rn×n is a constant matrix defined in Section 2.4. The plant parameterization (5.5.1) is of order 3n and has 2n unknown parameters. The state ω and signal η0 decay to zero exponentially fast with a rate that depends on Λc . Because Λc is arbitrary, it can be chosen so that η0 , ω go to zero faster than a certain given rate. Because φ1 (0) = φ2 (0) = 0,

288

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

the states φ1 , φ2 can be reproduced by the observer ˙ φˆ1 = Λc φˆ1 + lu, ˙ φˆ2 = Λc φˆ2 − ly,

φˆ1 (0) = 0 φˆ2 (0) = 0

(5.5.2)

which implies that φˆi (t) = φi (t), i = 1, 2 ∀t ≥ 0. The output of the observer is given by z0 = θ∗> φ y0 = θ∗> φˆ − λ> φˆ2 = θ∗> φ − λ> φ2

(5.5.3)

The state ω in (5.5.1) can not be reproduced exactly unless the initial condition x0 and therefore ω0 is known. Equation (5.5.2) and (5.5.3) describe the nonminimal state observer for ˆ = φ(t), ∀t ≥ 0, it follows the plant (5.3.1) when θ∗ is known. Because φ(t) 4

4

that the output observation errors ez = z − z0 , e0 = y − y0 satisfy ez = e0 = η0 = C0> eΛc t ω0 which implies that e0 , ez decay to zero exponentially fast. The eigenvalues of Λc can be regarded as the eigenvalues of the observer and can be assigned arbitrarily through the design of Λc . When θ∗ is unknown, the observer equation (5.5.2) remains the same but (5.5.3) becomes zˆ = yˆ + λ> φˆ2 yˆ = θ> φˆ − λ> φˆ2

(5.5.4)

ˆ is generated from where θ(t) is the estimate of θ∗ at time t and φ(t) = φ(t) ∗ (5.5.2). Because θ satisfies the parametric model z = y + λ> φ2 = θ∗> φ + η0

(5.5.5)

where z, φ are available for measurement and η0 is exponentially decaying to zero, the estimate θ(t) of θ∗ may be generated using (5.5.5) and the results of Chapter 4. As shown in Chapter 4, the exponentially decaying to zero term η0 does not affect the properties of the adaptive laws developed for η0 = 0 in (5.5.5). Therefore, for design purposes, we can assume that

5.5. NONMINIMAL ADAPTIVE OBSERVER

289

η0 ≡ 0 ∀t ≥ 0 and select any one of the adaptive laws from Tables 4.2, 4.3, and 4.5 to generate θ(t). In the analysis, we include η0 6= 0 and verify that its presence does not affect the stability properties and steady state behavior of the adaptive observer. As an example, let us use Table 4.2 and choose the gradient algorithm θ˙ = Γ²φ,

² = z − zˆ = y − yˆ

(5.5.6)

ˆ where Γ = Γ> > 0 and φ = φ. Equations (5.5.2), (5.5.4) and (5.5.6) form the adaptive observer and are summarized in Table 5.4 and shown in Figure 5.3. The stability properties of the adaptive observer in Table 5.4 are given by the following theorem. Theorem 5.5.1 The adaptive observer for the nonminimal plant representation (5.5.1) with the adaptive law based on the gradient algorithm or any other adaptive law from Tables 4.2, 4.3, and 4.5 that is based on the parametric model (5.5.5) guarantees that ˆ = φ(t) ∀t ≥ 0. (i) φ(t) (ii) All signals are u.b. (iii) The output observation error ²(t) = y(t) − yˆ(t) converges to zero as t → ∞. (iv) If u is sufficiently rich of order 2n, then θ(t) converges to θ∗ . The convergence of θ(t) to θ∗ is exponential for all the adaptive laws of Tables 4.2, 4.3, and 4.5 with the exception of the pure least squares where convergence is asymptotic. Proof (i) This proof follows directly from (5.5.1) and (5.5.2). (ii) Because φˆ = φ, the following properties of the adaptive laws can T be established using the results of Chapter 4: (a) ², θ ∈ L∞ ; (b) ², θ˙ ∈ L∞ L2 for the continuous time adaptive laws and |θk+1 − θk | ∈ `2 for the hybrid one. From ², θ, θ˙ ∈ L∞ , u ∈ L∞ , the stability of Λc and the stability of the plant, we have that all signals are u.b. (iii) For the continuous-time adaptive laws, we have that ² = z − zˆ = y + λ> φ2 − (ˆ y + λ> φ2 ) = y − yˆ We can verify that ²˙ ∈ L∞ which together with ² ∈ L2 imply that ²(t) → 0 as t → ∞.

290

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS Table 5.4 Adaptive observer (Realization 1)

Plant

φ˙ 1 = Λc φ1 + lu, φ1 (0) = 0 ˙ φ2 = Λc φ2 − ly, φ2 (0) = 0 ω˙ = Λc ω, ω(0) = ω0 η0 = C0> ω z = y + λ> φ2 = θ∗> φ + η0 y = θ∗> φ − λ> φ2 + η0

Observer

˙ φˆ1 = Λc φˆ1 + lu, ˙ φˆ2 = Λc φˆ2 − ly, zˆ = θ> φˆ yˆ = zˆ − λ> φˆ2

Adaptive law

ˆ θ˙ = Γ²φ, ² = z − zˆ,

Design variables

φˆ1 (0) = 0 φˆ2 (0) = 0

Γ = Γ> > 0 zˆ = θ> φˆ

Λc is a stable matrix; (Λc , l) is in the controller form; Λ(s) = det(sI − Λc ) = sn + λ> αn−1 (s)

For the hybrid adaptive law, we express ² as ² = θ∗> φ − θk> φ = −θ˜k> φ,

∀t ∈ [tk , tk+1 )

Because the hybrid adaptive law guarantees that (a) θ˜k ∈ l∞ , (b) |θ˜k+1 −θ˜k | ∈ l2 and |θ˜k+1 − θ˜k | → 0 as k → ∞, we can construct a continuous, piecewise linear function ˜ from θ˜k using linear interpolation that satisfies: (a) |θ˜ − θ˜k |, θ˜˙ ∈ L∞ T L2 and θ(t) ˜ − θ˜k (t)| → 0 as t → ∞. Therefore, we can write (b) |θ(t) ² = −θ˜k> φ = −θ˜> φ + (θ˜ − θ˜k )> φ

(5.5.7)

d ˜> (θ φ) = From ² ∈ L2 , |θ˜ − θ˜k | ∈ L2 and φ ∈ L∞ , we have θ˜> φ ∈ L2 . Because dt > ˙ ˙ d ˜> > ˙ ˜ ˜ ˜ ˜ ˙ θ φ + θ φ and θ, θ, φ, φ ∈ L∞ , we conclude that dt (θ φ) ∈ L∞ , which, together

5.5. NONMINIMAL ADAPTIVE OBSERVER

291 y

- Plant

u - αn−2 (s) - θ> 1 Λ(s) ˆ φ1 ¢¢ ¢

¢¸

+ - Σl AK− +6

yˆ

A

λ> ¾ ¢¸

θ2> θ1

¡¢

+ ² ? Σl − 6

¢ ¢

¾

φˆ2

θ2

−αn−2 (s) ¾ Λ(s) Adaptive Law ¾ ¾ (5.5.6)

φ

Figure 5.3 Adaptive observer using nonminimal Realization 1.

with θ˜> φ ∈ L2 , implies that θ˜> φ → 0 as t → ∞. Therefore, we have established that (θ˜ − θ˜k )> φ → 0 and θ˜> φ → 0 as t → ∞. Thus, using (5.5.7) for ², we have that ²(t) → 0 as t → ∞. (iv) The proof of this part follows directly from the properties of the adaptive laws developed in Chapter 4 and Theorem 5.2.4 by noticing that φ = H(s)u where H(s) is the same as in the proof of Theorem 5.2.4. 2

The reader can verify that the adaptive observer of Table 5.4 is the statespace representation of the parameter identifier given by (5.2.17), (5.2.18) in Section 5.2.3. This same identifier is shared by the adaptive Luenberger observer indicating that the state φ of the nonminimal state space representation of the plant is also part of the state of the adaptive Luenberger observer. In addition to φ, the adaptive Luenberger observer estimates the state of a minimal state space representation of the plant expressed in the observer canonical form at the expense of implementing an additional nth order state equation referred to as the observer equation in Table 5.1. The parameter identifiers of Section 5.2 can, therefore, be viewed as adaptive observers based on a nonminimal state representation of the plant.

292

5.5.2

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

Adaptive Observer Based on Realization 2

An alternative nonminimal state-space realization for the plant (5.3.1), developed in Chapter 2 and shown in Figure 2.5, is described by the state equations x ¯˙ 1 = −λ0 x ¯1 + θ¯∗> φ, x ¯1 (0) = 0 φ˙ 1 = Λc φ1 + lu, φ1 (0) = 0 φ˙ 2 = Λc φ2 − ly, φ2 (0) = 0 φ =

(5.5.8)

> > [u, φ> 1 , y, φ2 ]

ω˙ = Λc ω,

ω(0) = ω0

η0 = C0> ω y = x ¯ 1 + η0 where x ¯1 ∈ R; φi ∈ Rn−1 , i = 1, 2; θ¯∗ ∈ R2n is a vector of linear combinations of the unknown plant parameters a = [an−1 , an−2 , . . . , a0 ]> , b = [bn−1 , bn−2 , . . . , b0 ]> in the plant transfer function (5.3.7) as shown in Section 2.4; λ0 > 0 is a known scalar; Λc ∈ R(n−1)×(n−1) is a known stable (s) ∗ n−1 + q n−2 + . . . + q matrix; and (sI − Λc )−1 l = αΛn−2 ∗ (s) , where Λ (s) = s n−2 s 0 ∗ is a known Hurwitz polynomial. When θ is known, the states x ¯1 , φ1 , φ2 can be generated exactly by the observer ˆ x˙ o1 = −λ0 xo1 + θ¯∗> φ, xo1 (0) = 0 ˙ φˆ1 = Λc φˆ1 + lu, φˆ1 (0) = 0 ˙ φˆ2 = Λc φˆ2 − ly, φˆ2 (0) = 0

(5.5.9)

y0 = xo1 where xo1 , φˆi are the estimates of x ¯1 , φ respectively. As in Section 5.5.1, no attempt is made to generate an estimate of the state ω, because ω(t) → 0 as t → ∞ exponentially fast. The state observer (5.5.9) guarantees that xo1 (t) = x ¯1 (t), φˆ1 (t) = φ1 (t), φˆ2 (t) = φ2 (t) ∀t ≥ 0. The observation error 4

e0 = y − y0 satisfies

e0 = η0 = C0> eΛc t ω0

i.e., e0 (t) → 0 as t → ∞ with an exponential rate that depends on the matrix Λc that is chosen by the designer.

5.5. NONMINIMAL ADAPTIVE OBSERVER

293

When θ¯∗ is unknown, (5.5.9) motivates the adaptive observer ˆ x ˆ˙ 1 = −λ0 x ˆ1 + θ¯> φ, ˙ φˆ1 = Λc φˆ1 + lu, ˙ φˆ2 = Λc φˆ2 − ly,

x ˆ0 (0) = 0 φˆ1 (0) = 0 φˆ2 (0) = 0

(5.5.10)

yˆ = x ˆ1 ¯ where θ(t) is the estimate of θ¯∗ to be generated by an adaptive law and x ˆ1 is the estimate of x ¯1 . The adaptive law for θ¯ is developed using the SPR-Lyapunov design approach as follows: We define the observation error y˜ = y − yˆ = x ˜1 + η0 , where x ˜1 = x ¯1 − x ˆ1 , and use it to develop the error equation y˜˙ = −λ0 y˜ − θ˜> φ + C1> ω

(5.5.11)

where C1> = λ0 C0> + C0> Λc and θ˜ = θ¯ − θ¯∗ by using (5.5.8) and (5.5.9), and ˆ Except for the exponentially decaying to zero term C > ω, the fact that φ = φ. 1 equation (5.5.11) is in the appropriate form for applying the SPR-Lyapunov design approach. In the analysis below, we take care of the exponentially decaying term > C1 ω by choosing the Lyapunov function candidate V =

y˜2 θ˜> Γ−1 θ˜ + + βω > Pc ω 2 2

where Γ = Γ> > 0; Pc = Pc> > 0 satisfies the Lyapunov equation Pc Λc + Λ> c Pc = −I and β > 0 is a scalar to be selected. The time derivative V˙ of V along the solution of (5.5.11) is given by ˙ V˙ = −λ0 y˜2 − y˜θ˜> φ + y˜C1> ω + θ˜> Γ−1 θ˜ − βω > ω

(5.5.12)

As described earlier, if the adaptive law is chosen as ˙ θ˜ = θ¯˙ = Γ˜ yφ (5.5.12) becomes V˙ = −λ0 y˜2 − βω > ω + y˜C1> ω

(5.5.13)

294

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

If β is chosen as β > that

|C1 |2 2λ0 ,

then it can be shown by completing the squares λ0 β V˙ ≤ − y˜2 − ω > ω 2 2

which implies that V, θ, y˜ ∈ L∞ and y˜ ∈ L2 . Because from (5.5.11) y˜˙ ∈ L∞ it follows that y˜ → 0 as t → ∞, i.e., yˆ(t) → y(t) and x ˆ1 (t) → x ¯ (t) as t → ∞. i> 1 h > > ˆ ˆ ˆ converges to the Hence, the overall state of the observer X = x ˆ1 , φ , φ h

i>

1

2

> overall plant state X = x ¯1 , φ> as t → ∞. In addition θ¯˙ ∈ L2 ∩ L∞ 1 , φ2 ¯˙ and limt→∞ θ(t) = 0. ¯ to θ¯∗ depends on the properties of the input u. The convergence of θ(t) We can show by following exactly the same steps as in the previous sections that if u is sufficiently rich of order 2n, then φˆ = φ is P E, which, together ¯ converges to θ¯∗ exponentially fast. For with φ, φ˙ ∈ L∞ , implies that θ(t) φ˙ ∈ L∞ , we require, however, that u˙ ∈ L∞ . The main equations of the adaptive observer are summarized in Table 5.5 and the block diagram of the observer is shown in Figure 5.4 where θ¯ = [θ¯1> , θ¯2> ]> is partitioned into θ¯1 ∈ Rn , θ¯2 ∈ Rn .

Example 5.5.1 Consider the LTI plant y=

s2

b1 s + b0 u + a1 s + a0

(5.5.14)

where a1 , a0 > 0 and b1 , b0 are the unknown parameters. We first obtain the plant representation 2 by following the results and approach presented in Chapter 2. We choose Λ(s) = (s + λ0 )(s + λ) for some λ0 , λ > 0. It follows from (5.5.14) that · ¸ · ¸ s2 1 1 s s y = [b1 , b0 ] u − [a1 , a0 ] y 1 Λ(s) 1 Λ(s) Λ(s) Because

s2 Λ(s)

=1−

(λ0 +λ)s+λ0 λ Λ(s)

y = θ1∗>

we have

α1 (s) α1 (s) ¯ > α1 (s) y u − θ2∗> y+λ Λ(s) Λ(s) Λ(s)

(5.5.15)

¯ = [λ0 + λ, λ0 λ]> and α1 (s) = [s, 1]> . Because where θ1∗ = [b1 , b0 ]> , θ2∗ = [a1 , a0 ]> , λ Λ(s) = (s + λ0 )(s + λ), equation (5.5.15) implies that · ¸ 1 α1 (s) α1 (s) ¯ > α1 (s) y y= θ1∗> u − θ2∗> y+λ s + λ0 s+λ s+λ s+λ

5.5. NONMINIMAL ADAPTIVE OBSERVER

295

Table 5.5 Adaptive observer (Realization 2) x ¯˙ 1 = −λ0 x ¯1 + θ¯∗> φ, x ¯1 (0) = 0 ˙ φ1 = Λc φ1 + lu, φ1 (0) = 0 φ˙ 2 = Λc φ2 − ly, φ2 (0) = 0 ω˙ = Λc ω, ω(0) = ω0 > η0 = C0 ω y=x ¯ 1 + η0 > > where φ = [u, φ> 1 , y, φ2 ] n−1 φi ∈ R , i = 1, 2; x ¯1 ∈ R1

Plant

Observer

ˆ x ˆ˙ 1 = −λ0 x ˆ1 + θ¯> φ, x ˆ1 (0) = 0 ˙ˆ ˆ ˆ φ1 = Λc φ1 + lu, φ1 (0) = 0 ˙ ˆ ˆ φˆ2 (0) = 0 φ2 = Λc φ2 − ly, yˆ = x ˆ1 ˆ> > where φˆ = [u, φˆ> 1 , y, φ2 ] φˆi ∈ Rn−1 , i = 1, 2, x ˆ1 ∈ R1

Adaptive law

θ¯˙ = Γ˜ y φˆ , y˜ = y − yˆ

Design variables

Γ = Γ> > 0; Λc ∈ R(n−1)×(n−1) is any stable matrix, and λ0 > 0 is any scalar

Substituting for α1 (s) 1 = s+λ s+λ we obtain y

=

·

s 1

¸

· =

1 0

¸

1 + s+λ

·

−λ 1

¸

· 1 1 1 b1 u + (b0 − λb1 ) u − a1 y − (a0 − λa1 ) y s + λ0 s+λ s+λ ¸ 1 +(λ0 + λ)y − λ2 y s+λ

296

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS y

- Plant

u

- αn−2 (s) Λ(s)

φˆ1

- > ¯ - θ1

¢¢¸

¢¢

+ - Σl + 6

1 s + λ0

-

ˆ2 ¢¢¸ φ ¾ θ¯2> ¾

¢

+ y˜ ? Σl − ¡¢ yˆ 6

−αn−2 (s) ¾ Λ(s)

y

¢ ¢

θ¯2

θ¯1

Adaptive Law ¾ ¾ (5.5.13)

φ

Figure 5.4 Adaptive observer using nonminimal Realization 2. which implies that x ¯˙ 1 φ˙ 1 φ˙ 2 y

= = = =

−λ0 x ¯1 + θ¯∗> φ, x ¯1 (0) = 0 −λφ1 + u, φ1 (0) = 0 −λφ2 − y, φ2 (0) = 0 x ¯1

where φ = [u, φ1 , y, φ2 ]> , θ¯∗ = [b1 , b0 − λb1 , λ0 + λ − a1 , a0 − λa1 + λ2 ]> . Using Table 5.5, the adaptive observer for estimating x ¯1 , φ1 , φ2 and θ∗ is given by x ˆ˙ 1 ˙ φˆ1 ˙ φˆ2 yˆ θ¯˙

ˆ = −λ0 x ˆ1 + θ¯> φ,

x ˆ1 (0) = 0

= −λφˆ1 + u,

φˆ1 (0) = 0

= −λφˆ2 − y, = x ˆ1 ˆ − yˆ) = Γφ(y

φˆ2 (0) = 0

where φˆ = [u, φˆ1 , y, φˆ2 ]> and Γ = Γ> > 0. If in addition to θ¯∗ , we like to estimate θ∗ = [b1 , b0 , a1 , a0 ]> , we use the relationships ˆb1 ˆb0

= θ¯1 = θ¯2 + λθ¯1

a ˆ1

= −θ¯3 + λ0 + λ = θ¯4 − λθ¯3 + λλ0

a ˆ0

5.6. PARAMETER CONVERGENCE PROOFS

297

where θ¯i , i = 1, 2, 3, 4 are the elements of θ¯ and ˆbi , a ˆi , i = 1, 2 are the estimates of bi , ai , i = 0, 1, respectively. For parameter convergence we choose u = 6 sin 2.6t + 8 sin 4.2t

2

which is sufficiently rich of order 4.

5.6

Parameter Convergence Proofs

In this section we present all the lengthy proofs of theorems dealing with convergence of the estimated parameters.

5.6.1

Useful Lemmas

The following lemmas are used in the proofs of several theorems to follow: Lemma 5.6.1 If the autocovariance of a function x : R+ 7→ Rn defined as 4

1 T →∞ T

Z

t0 +T

Rx (t) = lim

x(τ )x> (t + τ )dτ

(5.6.1)

t0

exists and is uniform with respect to t0 , then x is PE if and only if Rx (0) is positive definite. Proof If: The definition of the autocovariance Rx (0) implies that there exists a T0 > 0 such that Z t0 +T0 1 1 3 Rx (0) ≤ x(τ )x> (τ )dτ ≤ Rx (0), ∀t ≥ 0 2 T0 t0 2 If Rx (0) is positive definite, there exist α1 , α2 > 0 such that α1 I ≤ Rx (0) ≤ α2 I. Therefore, Z t0 +T0 1 1 3 α1 I ≤ x(τ )x> (τ )dτ ≤ α2 I 2 T0 t0 2 for all t0 ≥ 0 and thus x is PE. Only if: If x is PE, then there exist constants α0 , T1 > 0 such that Z t

t+T1

x(τ )x> (τ )dτ ≥ α0 T1 I

298

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

for all t ≥ 0. For any T > T1 , we can write Z

t0 +T

x(τ )x> (τ )dτ

k−1 X Z t0 +(i+1)T1

=

t0

i=0

Z x(τ )x> (τ )dτ +

t0 +iT1

t0 +T

x(τ )x> (τ )dτ

t0 +kT1

≥ kα0 T1 I where k is the largest integer that satisfies k ≤ T /T1 , i.e., kT1 ≤ T < (k + 1)T1 . Therefore, we have Z kT1 1 t+T x(τ )x> (τ )dτ ≥ α0 I T t T For k ≥ 2, we have

kT1 T

=

(k+1)T1 T

1 T

Z

T1 T

−

t0 +T

≥1−

T1 T

≥ 12 , thus,

x(τ )x> (τ )dτ ≥

t0

and

Z

1 Rx (0) = lim T →∞ T

t0 +T

α0 I 2

x(τ )x> (τ )dτ ≥

t0

α0 I 2

which implies that Rx (0) is positive definite.

2

Lemma 5.6.2 Consider the system y = H(s)u where H(s) is a strictly proper transfer function matrix of dimension m × n with stable poles and real impulse response h(t). If u is stationary, with autocovariance Ru (t), then y is stationary, with autocovariance Z ∞Z ∞ Ry (t) = h(τ1 )Ru (t + τ1 − τ2 )h> (τ2 )dτ1 dτ2 −∞

−∞

and spectral distribution Sy (ω) = H(−jω)Su (ω)H > (jω) Proof See [201]. Lemma 5.6.3 Consider the system described by · ¸ · ¸· ¸ x˙ 1 x1 A −F > (t) = x˙ 2 x2 P1 F (t)P2 0

(5.6.2)

5.6. PARAMETER CONVERGENCE PROOF

299

where x1 ∈ Rn1 , x2 ∈ Rrn1 for some integer r, n1 ≥ 1, A, P1 , P2 are constant matrices and F (t) is of the form   z1 In1  z2 In1    F (t) =  .  ∈ Rrn1 ×n1  ..  zr In1 where zi , i = 1, 2, . . . , r are the elements of the vector z ∈ Rr . Suppose that z is PE and there exists a matrix P0 > 0 such that > P˙0 + A> 0 P0 + P0 A0 + C0 C0 ≤ 0

where

· A0 =

A P1 F (t)P2

−F > (t) 0

(5.6.3)

¸ , C0> = [In1 , 0]

Then the equilibrium x1e = 0, x2e = 0 of (5.6.2) is e.s. in the large. Proof Consider the system (5.6.2) that we express as x˙ = A0 (t)x y = C0> x = x1

(5.6.4)

> > where x = [x> 1 , x2 ] . We first show that (C0 , A0 ) is UCO by establishing that > (C0 , A0 + KC0 ) is UCO for some K ∈ L∞ which according to Lemma 4.8.1 implies that (C0 , A0 ) is UCO. We choose ¸ · −γIn1 − A K= −P1 F (t)P2

for some γ > 0 and consider the following system associated with (C0 , A0 + KC0> ): ¸ · ¸· ¸ · Y˙ 1 Y1 −γIn1 −F > (t) = Y2 0 Y˙ 2 · 0 ¸ (5.6.5) Y1 y1 = [In1 0] Y2 According to Lemma 4.8.4, the system (5.6.5) is UCO if Ff (t) = satisfies αIrn1

1 ≤ T

Z

t+T t

1 F (t) s+γ

Ff (τ )Ff> (τ )dτ ≤ βIrn1 , ∀t ≥ 0

(5.6.6)

300

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

for some constants α, β, T > 0. We prove (5.6.6) by first showing that F (t) satisfies Z 0 0 1 t+T α Irn1 ≤ F (τ )F > (τ )dτ ≤ β Irn1 , ∀t ≥ 0 T t 0

0

for some constants α , β as follows: Using a linear transformation, we can express F (t) as F (t) = F0 Z(t) where F0 ∈ Rrn1 ×rn1 is a constant matrix of full rank, Z ∈ Rrn1 ×n1 is a block 4 diagonal matrix defined as Z = diag{z, z, . . . , z }, i.e., | {z } n1

           Z=          

z1 z2 .. .

0 0 .. .

··· ···

zn1 0 .. .

0 z1 .. .

···

0 .. .

zn1 .. .

0 .. .

0 .. .

···

0

0

···

0 0 .. .



     0   0   ..  .   0   ..  .   z1   ..  .  zn1

Therefore, F F > = F0 ZZ > F0> and ZZ > = diag{zz > , zz > , . . . , zz > } | {z } n1

Because z is PE, we have 1 T

α 1 Ir ≤

Z

t+T

zz > dτ ≤ α2 Ir ,

∀t ≥ 0

t

for some α1 , α2 , T > 0. Therefore, Z 1 t+T α1 Irn1 ≤ Z(τ )Z > (τ )dτ ≤ α2 Irn1 , T t

∀t ≥ 0

which implies that α1 F0 F0>

1 ≤ T

Z t

t+T

F (τ )F > (τ )dτ ≤ α2 F0 F0> ,

∀t ≥ 0

5.6. PARAMETER CONVERGENCE PROOF

301

Because F0 is of full rank, we have β1 Irn1 ≤ F0 F0> ≤ β2 Irn1 for some constants β1 , β2 > 0. Hence, Z 0 0 1 t+T α Irn1 ≤ F (τ )F > (τ )dτ ≤ β Irn1 , ∀t ≥ 0 T t 0

(5.6.7)

0

where β = α2 β2 , α = α1 β1 . Following the same arguments used in proving Lemma 4.8.3 (iv), one can show (see Problem 5.18) that (5.6.7) implies (5.6.6). Because all the conditions in Lemma 4.8.4 are satisfied, we conclude, by applying Lemma 4.8.4 that (5.6.5) is UCO, which in turn implies that (5.6.2) is UCO. Therefore, it follows directly from Theorem 3.4.8 and (5.6.3) that the equilibrium x1e = 0, x2e = 0 of (5.6.2) is e.s. in the large. 2

5.6.2

Proof of Theorem 5.2.1

According to Lemma 5.6.1, Theorem 5.2.1 can be proved if we establish that Rφ (0) is positive definite if and only if u is sufficiently rich of order n. If: We will show the result by contradiction. Because u is stationary and Rφ (0) is uniform with respect to t, we take t = 0 and obtain [186] Z Z ∞ 1 T 1 Rφ (0) = lim φ(τ )φ> (τ )dτ = Sφ (ω)dω (5.6.8) T →∞ T 0 2π −∞ where Sφ (ω) is the spectral distribution of φ. From Lemma 5.6.2, we have Sφ (ω) = H(−jω)Su (ω)H > (jω)

(5.6.9)

Using the condition that u is sufficiently rich of order n, i.e., u has spectral lines at n points, we can express Su (ω) as Su (ω) =

n X

fu (ωi )δ(ω − ωi )

(5.6.10)

i=1

where fu (ωi ) > 0. Using (5.6.9) and (5.6.10) in (5.6.8), we obtain n 1 X Rφ (0) = fu (ωi )H(−jωi )H > (jωi ) 2π i=1

Suppose that Rφ (0) is not positive definite, then there exists x ∈ Rn with x 6= 0 such that n X x> Rφ (0)x = fu (ωi )x> H(−jωi )H > (jωi )x = 0 (5.6.11) i=1

302

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

Because fu (ωi ) > 0 and each term under the summation is nonnegative, (5.6.11) can be true only if: x> H(−jωi )H > (jωi )x = 0, i = 1, 2, . . . , n or equivalently x> H(−jωi ) = 0,

i = 1, 2, . . . , n

(5.6.12)

However, (5.6.12) implies that {H(jω1 ), H(jω2 ), . . . , H(jωn )} are linearly dependent, which contradicts with the condition that H(jω1 ), . . . H(jωn ) are linearly independent for all ω1 , . . . , ωn . Hence, Rφ (0) is positive definite. Only if: We also prove this by contradiction. Assume that Rφ (0) is positive definite but u is sufficiently rich of order r < n, then we can express Rφ (0) as Rφ (0) =

r 1 X fu (ωi )H(−jωi )H > (−jωi ) 2π i=1

where fu (ωi ) > 0. Note that the right hand side is the sum of r − dyads, and the rank of Rφ (0) can be at most r < n, which contradicts with the assumption that Rφ (0) is positive definite. 2

5.6.3

Proof of Theorem 5.2.2

We first consider the series-parallel scheme (5.2.11). From (5.2.10), (5.2.11) we obtain the error equations ˜ − Ax ˜ ²˙1 = Am ²1 − Bu ˙ ˙ ˜ = γ²1 u A˜ = γ²1 x> , B

(5.6.13)

4 4 ˜= ˆ − B and B ˜ ∈ Rn×1 , A˜ ∈ An×n . For simplicity, let us take where A˜ = Aˆ − A, B B γ1 = γ2 = γ. The parameter error A˜ is in the matrix form, which we rewrite in the familiar vector form to apply the stability theorems of Chapter 3 directly. Defining the vector 4 n(n+1) ˜> > θ˜ = [˜ a> ˜> ˜> 1 ,a 2 ,...,a n,B ] ∈ R

˜ we can write where a ˜i is the ith column of A,  x1  x2  ˜ + Bu ˜ = [˜ Ax a1 , a ˜2 , . . . , a ˜n ]  .  .. xn

   ˜  + Bu = F > (t)θ˜ 

5.6. PARAMETER CONVERGENCE PROOF

303

4 ˜˙ = γ²1 u, where F > (t) = [x1 In , x2 In , . . . , xn In , uIn ] ∈ Rn×n(n+1) . Because a ˜˙ i = γ²1 xi , B ˙˜ B ˜˙ can be rewritten as the matrix differential equations for A,

˙ θ˜ = γF (t)²1 Therefore, (5.6.13) is equivalent to ( ²˙1 = Am ²1 − F > (t)θ˜ ˙ θ˜ = γF (t)²1

(5.6.14)

which is in the form of (5.6.2). To apply Lemma 5.6.3, we need to verify that all the conditions stated in the lemma are satisfied by (5.6.14). We first prove that there exists a constant matrix P0 > 0 such that > A> 0 P0 + P0 A0 = −C0 C0

where A0 =

·

A −F > (t) γF (t) 0

¸ ∈ R(n+n(n+1))×(n+n(n+1)) , C0> = [In , 0] ∈ Rn+n(n+1)

In Section 4.2.3, we have shown that the time derivative of the Lyapunov function ) ) ( ( ˜ >P B ˜ A˜> P A˜ B > V = ²1 P ²1 + tr + tr γ1 γ2 (with γ1 = γ2 = γ) satisfies where P satisfies

A> mP

V˙ = −²> 1 ²1

(5.6.15)

+ P Am = −In . Note that o n = tr A˜> P A˜

n X

n o ˜ >P B ˜ tr B =

i=1

a ˜> ˜i i Pa

˜ >P B ˜ B

˜ ∈ Rn×1 . We can write where the second equality is true because B V =

²> 1 P ²1

n 1 X > 1 ˜> ˜ + a ˜ Pa ˜i + B P B = x> P0 x γ1 i=1 i γ2

4 ˜> > where x = [²> 1 , θ ] and P0 is a block diagonal matrix defined as 4

P0 = diag{P, γ1−1 P, . . . , γ1−1 P , γ2−1 P } ∈ R(n+n(n+1))×(n+n(n+1)) | {z } n−times

304

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

Hence, (5.6.15) implies that > > V˙ = x> (P0 A0 + A> 0 P0 )x = −x C0 C0 x

or equivalently

> ˙ P˙0 + P0 A0 + A> 0 P0 = −C0 C0 , P0 = 0

(5.6.16)

4

Next, we show that z = [x1 , x2 , . . . , xn , u]> is PE. We write · ¸ (sI − A)−1 B z = H(s)u, H(s) = 1 If we can show that H(jω1 ), H(jω2 ), . . . , H(jωn+1 ) are linearly independent for any ω1 , ω2 , . . . , ωn+1 , then it follows immediately from Theorem 5.2.1 that z is PE if and only if u is sufficiently rich of order n + 1. Let a(s) = det(sI − A) = sn + an−1 sn−1 + . . . + a1 s + a0 . We can verify using matrix manipulations that the matrix (sI − A)−1 can be expressed as (sI − A)−1

=

1 © n−1 Is + (A + an−1 I)sn−2 + (A2 + an−1 A + an−2 I)sn−3 a(s) ª + . . . + (An−1 + an−1 An−2 + . . . + a1 I) (5.6.17)

Defining b1 = B, b2 = (A+an−1 I)B, b3 = (A2+an−1 A+an−2 I)B, . . . , bn = (An−1+. . .+a1I)B L = [b1 , b2 , . . . , bn ] ∈ Rn×n and then using (5.6.17), H(s) can be conveniently expressed as   n−1   s   sn−2         ..   1  L .   H(s) =       s   a(s)     1 a(s) To explore the linear dependency of H(jωi ), we define H(jωn+1 )]. Using the expression (5.6.18) for H(s), we  (jω1 )n−1 (jω2 )n−1  (jω1 )n−2 (jω2 )n−2 · ¸  .. .. L 0n×1  . . ¯ H =  01×n 1  jω jω 1 2   1 1 a(jω1 ) a(jω2 )

(5.6.18)

4 ¯ = H [H(jω1 ) , H(jω2 ), . . ., have  · · · (jωn+1 )n−1 · · · (jωn+1 )n−2    ..  .   ··· jωn+1   ··· 1 ··· a(jωn+1 )

5.6. PARAMETER CONVERGENCE PROOF    ×  

1 a(jω1 )

0 .. . 0

0 1 a(jω2 )

···

··· ··· .. . 0

0 0 0

305      

1 a(jωn+1 )

From the assumption that (A, B) is controllable, we conclude that the matrix L ¯ has rank of n + 1 if and only if the matrix V1 is of full rank. Thus the matrix H defined as   (jω1 )n−1 (jω2 )n−1 · · · (jωn+1 )n−1  (jω1 )n−2 (jω2 )n−2 · · · (jωn+1 )n−2      .. .. .. 4   . . . V1 =     jω1 jω2 ··· jωn+1     1 1 ··· 1 a(jω1 ) a(jω2 ) ··· a(jωn+1 ) has rank of n + 1. Using linear transformations (row operations), we can show that V1 is equivalent to the following Vandermonde matrix [62]:      V =   

(jω1 )n (jω1 )n−1 (jω1 )n−2 .. .

(jω2 )n (jω2 )n−1 (jω2 )n−2 .. .

··· ··· ···

(jωn+1 )n (jωn+1 )n−1 (jωn+1 )n−2 .. .

jω1 1

jω2 1

··· ···

jωn+1 1

Because det(V ) =

Y

        

(jωi − jωk )

1≤i
V is of full rank for any ωi with ωi 6= ωk i, k = 1, . . . n + 1. This leads to ¯ have rank of n + 1, which implies that the conclusion that V1 and, therefore, H H(jω1 ), H(jω2 ), . . . , H(jωn+1 ) are linearly independent for any ω1 , ω2 , . . . ωn+1 . It then follows immediately from Theorem 5.2.1 that z is PE. Because we have shown that all the conditions of Lemma 5.6.3 are satisfied by (5.6.14), we can conclude that xe = 0 of (5.6.14) is e.s. in the large, i.e., ²1 , θ˜ → 0 ˆ → B exponentially fast as t → ∞, exponentially fast as t → ∞. Thus, Aˆ → A, B and the proof of Theorem 5.2.2 for the series-parallel scheme is complete. 4

For the parallel scheme, we only need to establish that zˆ = [ˆ x> , u]> is PE. The rest of the proof follows by using exactly the same arguments and procedure as in the case of the series-parallel scheme. Because x is the state of the plant and it is independent of the identification scheme, it follows from the previous analysis that z is PE under the conditions given

306

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

in Theorem 5.2.2, i.e., (A, B) controllable and u sufficiently rich of order n+1. From the definition of ²1 , we have · ¸ 0n×1 ²1 x ˆ = x − ²1 , zˆ = z − 1 thus the PE property of zˆ follows immediately from Lemma 4.8.3 by using ²1 ∈ L2 and z being PE. 2

5.6.4

Proof of Theorem 5.2.3

We consider the proof for the series-parallel scheme. The proof for the parallel scheme follows by using the same arguments used in Section 5.6.3 in the proof of Theorem 5.2.2 for the parallel scheme. Following the same procedure used in proving Theorem 5.2.2, we can write the differential equations ²˙1 = ˙ Aˆ = ˆ˙ = B

ˆ + (B − B)u ˆ Am ²1 + (A − A)x γ²1 x> γ²1 u>

in the vector form as ²˙1 = ˙ θ˜ =

Am ²1 − F > (t)θ˜ F (t)²1

(5.6.19)

4

˜> > and a ˜> ˜i , ˜bi denotes the ith column of ˜> , ˜b> ˜> where θ˜ = [˜ a> 1 , b2 , . . . , bq ] 2 ,...,a 1 ,a 4 ˜ B, ˜ respectively, F > (t) = [x1 In , x2 In , . . . , xn In , u1 In , u2 In , . . . , uq In ]. Following A, exactly the same arguments used in the proof for Theorem 5.2.2, we complete the > proof by showing (i) there exists a matrix P0 > 0 such that A> 0 P0 +P0 A0 = −C0 C0 , 4

where A0 , C0 are defined the same way as in Section 5.6.3 and (ii) z = [x1 , x2 , . . ., xn , u1 , u2 , . . ., uq ]> is PE. The proof for (i) is the same as that in Section 5.6.3. We prove (ii) by showing that the autocovariance of z, Rz (0) is positive definite as follows: We express z as · z=

(sI − A)−1 B Iq

¸ u=

¸ q · X (sI − A)−1 bi ui ei i=1

where Iq ∈ Rq×q is the identity matrix and bi , ei denote the ith column of B, Iq , respectively. Assuming that ui , i = 1, . . . , q are stationary and uncorrelated, the

5.6. PARAMETER CONVERGENCE PROOF

307

autocovariance of z can be calculated as q Z 1 X ∞ Rz (0) = Hi (−jω)Sui (ω)Hi> (jω) 2π i=1 −∞ where

·

(sI − A)−1 bi ei

Hi (s) =

¸

and Sui (ω) is the spectral distribution of ui . Using the assumption that ui is sufficiently rich of order n + 1, we have Sui (ω) =

n+1 X

fui (ωik )δ(ω − ωik )

k=1

and 1 2π

Z

∞

−∞

Hi (−jω)Sui (ω)Hi> (jω) =

n+1 1 X fui (ωik )Hi (−jωik )Hi> (jωik ) 2π k=1

where fui (ωik ) > 0. Therefore, Rz (0) =

q n+1 1 XX fui (ωik )Hi (−jωik )Hi> (jωik ) 2π i=1

(5.6.20)

k=1

Let us now consider the solution of the quadratic equation x> Rz (0)x = 0,

x ∈ Rn+q

Because each term under the summation of the right-hand side of (5.6.20) is semipositive definite, x>Rz (0)x = 0 is true if and only if fui (ωik )x> Hi (−jωik )Hi> (jωik )x = 0,

i = 1, 2, . . . q k = 1, 2, . . . n + 1

or equivalently i = 1, 2, . . . q k = 1, 2, . . . n + 1

Hi> (jωik )x = 0, · Because Hi (s) =

1 a(s)

adj(aI − A)bi a(s)ei

(5.6.21)

¸ where a(s) = det(sI −A), (5.6.21) is equiv-

alent to: ¯ i> (jωik )x = 0, H

4 ¯ i (s) = H a(s)Hi (s),

i = 1, 2, . . . q k = 1, 2, . . . n + 1

(5.6.22)

308

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS

¯ i is a polynomial of order at most equal to n, we Noting that each element in H 4 > ¯ find that gi (s) = Hi (s)x is a polynomial of order at most equal to n. Therefore, (5.6.22) implies that the polynomial gi (s) vanishes at n + 1 points, which, in turn, implies that gi (s) ≡ 0 for all s ∈ C. Thus, we have ¯ i> (s)x ≡ 0, H

i = 1, 2, . . . q

for all s. Equation (5.6.23) can be written in the matrix form · ¸> adj(sI − A)B x ≡ 0q a(s)Iq

(5.6.23)

(5.6.24)

where 0q ∈ Rq is a column vector with all elements equal to zero. Let X = [x1 , . . . , xn ] ∈ Rn , Y = [xn+1 , . . . , xn+q ] ∈ Rq , i.e., x = [X > , Y > ]> . Then (5.6.24) can be expressed as (adj(sI − A)B)> X + a(s)Y = 0q (5.6.25) Consider the following expressions for adj(sI − A)B and a(s): = Bsn−1 + (AB + an−1 B)sn−2 + (A2 B + an−1 AB + an−2 B)sn−3 + . . . + (An−1 B + an−1 An−2 B + . . . + a1 B) a(s) = sn + an−1 sn−1 + . . . + a1 s + a0

adj(sI − A)B

and equating the coefficients of si on both sides of equation (5.6.25), we find that X, Y must satisfy the following algebraic equations:  Y = 0q    B>X + a   n−1 Y = 0q  (AB + an−1 B)> X + an−2 Y = 0q  ..    .   (An−1 + an−1 An−2 B + . . . + a1 B)> X + a0 Y = 0q or equivalently: Y = 0q

and

(B, AB, . . . , An−1 B)> X = 0nq

(5.6.26)

where 0nq is a zero column-vector of dimension nq. Because (A, B) is controllable, the matrix (B, AB, . . . An−1 B) is of full rank; therefore, (5.6.26) is true if and only if X = 0, Y = 0, i.e. x = 0. Thus, we have proved that x> Rz (0)x = 0 if and only if x = 0, which implies that Rz (0) is positive definite. Then it follows from Lemma 5.6.1 that z is PE. Using exactly the same arguments as used in proving Theorem 5.2.2, we con˜ converge to zero exponentially fast as t → ∞. clude from z being PE that ²1 (t), θ(t) ˆ ˆ From the definition of θ˜ we have that A(t) → A, B(t) → B exponentially fast as t → ∞ and the proof is complete. 2

5.6. PARAMETER CONVERGENCE PROOF

5.6.5

309

Proof of Theorem 5.2.5

Let us define

˜ ²¯(t) = θ˜> (t)Rφ (0)θ(t)

˜ = θ(t) − θ∗ . We will show that ²¯(t) → 0 as t → ∞, i.e., for any given where θ(t) 0 0 ² > 0, there exists a t1 ≥ 0 such that for all t ≥ t1 , ²¯(t) < ² . We express ²¯ as Z 1 t1 +T ˜ ²¯(t) = θ˜> (t) φ(τ )φ> (τ )dτ θ(t) T t1 Ã ! Z 1 t1 +T > > ˜ ˜ +θ (t) Rφ (0) − φ(τ )φ (τ )dτ θ(t) T t1 Z Z o 1 t1 +T ˜> 1 t1 +T n ˜> = (θ (τ )φ(τ ))2 dτ + (θ (t)φ(τ ))2 − (θ˜> (τ )φ(τ ))2 dτ T t1 T t1 ! Ã Z t1 +T 1 > > ˜ ˜ φ(τ )φ (τ )dτ θ(t) +θ (t) Rφ (0) − T t1 4

=

²¯1 (t) + ²¯2 (t) + ²¯3 (t)

where t1 , T are arbitrary at this point and will be specified later. We evaluate each term on the right-hand side of the above equation separately. 0 ˜> Because q ²(t) = θ (t)φ(t) → 0 as t → ∞, there exists a t1 ≥ 0 such that |²(t)| <

²0 3

0

0

for t ≥ t1 . Choosing t1 ≥ t1 , we have |¯ ²1 (t)| =

1 T

Z

t1 +T

0

²2 (τ )dτ <

t1

² 3

(5.6.27)

For ²¯2 (t), we have ²¯2 (t)

= =

1 T 1 T

Z

t1 +T

t1

Z

t1 +T

t1

˜ − θ(τ ˜ ))> φ(τ )φ> (τ )(θ(t) ˜ + θ(τ ˜ ))dτ (θ(t) µ Z −

τ

˜˙ θ(σ)dσ

¶>

˜ + θ(τ ˜ ))dτ φ(τ )φ> (τ )(θ(t)

t

˙ Using the property of the adaptive law that θ˜ → 0 as t → ∞, for any given 0 0 0 0 2² ˜˙ T, ² we can find a t2 ≥ 0 such that for all t ≥ t2 , |θ(t)| ≤ 3KT , where K = ˜ + θ(τ ˜ ))| is a finite constant because of the fact that both θ˜ supt,τ |φ(τ )φ> (τ )(θ(t) 0 and φ are uniformly bounded signals. Choosing t1 ≥ t2 , we have Z 0 0 1 t1 +T 2² (τ − t) ² > ˜ ˜ |¯ ²2 (t)| ≤ |φ(τ )φ (τ )(θ(t) + θ(τ ))|dτ < (5.6.28) T t1 3KT 3

310

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS 0

for all t ≥ t2 . Because the autocovariance Rφ (0) exists, there exists T0 ≥ 0 such that for all T ≥ T0 , ° ° Z 0 ° ° 1 t1 +T ² ° ° > φ(τ )φ (τ )dτ ° < °Rφ (0) − ° ° 3K 0 T t1 0 ˜ 2 is a finite constant. Therefore, where K = supt |θ(t)| 0

|¯ ²3 | <

² 3

(5.6.29) 0

0

4

Combining (5.6.27) to (5.6.29), we have that for t ≥ max{t1 , t2 } = t1 , T ≥ T0 , ˜ <² ²¯(t) = θ˜> (t)Rφ (0)θ(t)

0

which, in turn, implies that ²¯; therefore, Rφ (0)θ˜ converges to zero as t → ∞.

5.7

2

Problems

5.1 Show that

· φ=

A sin(t + ϕ) sin(t)

¸

where A, ϕ are nonzero constants is PE in R2 . 5.2 Consider the following plant y=

b1 s u (s + 1)2

where b1 is the only unknown parameter. Is u = c0 (constant ) 6= 0 sufficiently rich for identifying b1 ? Explain. Design a parameter identifier to identify b1 from the measurements of u, y. Simulate your scheme on a digital computer for b1 = 5. 5.3 Consider the plant y=

b2 s2 + b0 u (s + 2)3

where b2 , b0 are the only unknown parameters. Is u = sin t sufficiently rich for identifying b2 , b0 where b2 , b0 can be any number in R. Explain. Design a parameter identifier to identify b2 , b0 . Simulate your scheme for a) b2 = 1, b0 = 1 and b) b2 = 3, b0 = 5.

5.7. PROBLEMS

311

5.4 Consider the second order stable plant · ¸ · ¸ a1 1 b1 x˙ = x+ u a0 0 1 where a1 , a0 , b1 are the only unknown parameters and u ∈ L∞ . (a) Design a parameter identifier to estimate the unknown parameters. (b) Choose u with the least number of frequencies that guarantees parameter convergence. (c) Simulate your scheme for a1 = −2, a0 = −5, b1 = 8. 5.5 Simulate the series-parallel identifier in Example 5.2.2. Repeat Example 5.2.2 by designing and simulating a parallel identifier. In simulations use numerical values for A, B of your choice. 5.6 Perform the simulations requested in (i) and (ii) of Example 5.2.3 when b0 = −2, a1 = 2.8, a0 = 5.6. Comment on your results. 5.7 Repeat Problem 5.6 when the pure least-squares algorithm in Example 5.2.3 is replaced with the least-squares with covariance resetting algorithm. 5.8 Repeat Problem 5.6 when the pure least-squares algorithm in Example 5.2.2 is replaced with the (a) Integral algorithm (b) Hybrid adaptive law. 5.9 Design an adaptive Luenberger observer for the plant · ¸ · ¸ −a1 1 b1 x˙ = x+ u −a0 0 1 y = [1, 0]x where a1 , a0 > 0 and b1 6= 0 are the only unknown parameters using the following adaptive laws for on-line estimation: (a) Integral algorithm (b) Pure least-squares (c) Hybrid adaptive law In each case present the complete stability proof. Simulate the adaptive observers with inputs u of your choice. For simulation purposes assume that the unknown parameters have the following values: a1 = 2.5, a0 = 3.6, b1 = 4. 5.10 Repeat Problem 5.9 by designing an adaptive observer with auxiliary input. 5.11 Consider the LTI plant y=

s2

b0 u + a1 s + a2

312

CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS where (a) (b) (c)

b0 6= 0, and a1 , a2 > 0. Represent the plant in the following forms: Observable form Nonminimal Realization 1 Nonminimal Realization 2

5.12 Design an adaptive observer using an integral adaptive law for the plant of Problem 5.11 represented in the observer form. 5.13 Repeat Problem 5.12 for the same plant in the nonminimal Realization 1. 5.14 Design an adaptive observer for the plant of Problem 5.11 expressed in the nonminimal Realization 2. 5.15 Consider the following plant: y = W0 (s)G(s)u where W0 (s) is a known proper transfer function with stable poles and G(s) is a strictly proper transfer function of order n with stable poles but unknown coefficients. (a) Design a parameter identifier to identify the coefficients of G(s). (b) Design an adaptive observer to estimate the states of a minimal realization of the plant. 5.16 Prove that there exists a signal vector v ∈ Rn available for measurement for which the system given by (5.4.2) becomes y˜ = −θ˜> φ ¸ · u ∈ R2n and H(s) is a known transfer matrix. where φ = H(s) y 5.17 Use the result of Problem 5.16 to develop an adaptive observer with auxiliary input that employs a least-squares algorithm as a parameter estimator. 5.18 Let F (t) : R 7→ Rn×m and F, F˙ ∈ L∞ . If there exist positive constants k1 , k2 , T0 such that Z t+T0 1 k 1 In ≤ F (τ )F > (τ )dτ ≤ k2 In T0 t for any t ≥ 0, where In is the identity matrix of dimension n, show that if Ff =

b F s+a 0

0

0

with a > 0, b 6= 0, then there exist positive constants k1 , k2 , T0 such that Z t+T00 0 0 1 k1 In ≤ 0 Ff (τ )Ff> (τ )dτ ≤ k2 In T0 t for any t ≥ 0.

Chapter 6

Model Reference Adaptive Control 6.1

Introduction

Model reference adaptive control (MRAC) is one of the main approaches to adaptive control. The basic structure of a MRAC scheme is shown in Figure 6.1. The reference model is chosen to generate the desired trajectory, 4 ym , that the plant output yp has to follow. The tracking error e1 = yp − ym represents the deviation of the plant output from the desired trajectory. The closed-loop plant is made up of an ordinary feedback control law that contains the plant and a controller C(θ) and an adjustment mechanism that generates the controller parameter estimates θ(t) on-line. The purpose of this chapter is to design the controller and parameter adjustment mechanism so that all signals in the closed-loop plant are bounded and the plant output yp tracks ym as close as possible. MRAC schemes can be characterized as direct or indirect and with normalized or unnormalized adaptive laws. In direct MRAC, the parameter vector θ of the controller C(θ) is updated directly by an adaptive law, whereas in indirect MRAC θ is calculated at each time t by solving a certain algebraic equation that relates θ with the on-line estimates of the plant parameters. In both direct and indirect MRAC with normalized adaptive laws, the form of C(θ), motivated from the known parameter case, is kept unchanged. The controller C(θ) is combined with an adaptive law (or an adaptive law and 313

314

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

-

µ ¡ - Controller C(θ)

r

ym

Reference Model

up-

− ? e1 Σl + 6

•

Plant

yp

¡

¡

¡

¡

θ

up Adjustment ¾ ¾ Mechanism ¾e1

yp

Figure 6.1 General structure of MRAC scheme. an algebraic equation in the indirect case) that is developed independently by following the techniques of Chapter 4. This design procedure allows the use of a wide class of adaptive laws that includes gradient, least-squares and those based on the SPR-Lyapunov design approach. On the other hand, in the case of MRAC schemes with unnormalized adaptive laws, C(θ) is modified to lead to an error equation whose form allows the use of the SPRLyapunov design approach for generating the adaptive law. In this case, the design of C(θ) and adaptive law is more complicated in both the direct and indirect case, but the analysis is much simpler and follows from a consideration of a single Lyapunov-like function. The chapter is organized as follows: In Section 6.2, we use several examples to illustrate the design and analysis of a class of simple direct MRAC schemes with unnormalized adaptive laws. These examples are used to motivate the more general and complicated designs treated in the rest of the chapter. In Section 6.3 we define the model reference control (MRC) problem for SISO plants and solve it for the case of known plant parameters. The control law developed in this section is used in the rest of the chapter to form MRAC schemes in the unknown parameter case. The design of direct MRAC schemes with unnormalized adaptive laws is treated in Section 6.4 for plants with relative degree n∗ = 1, 2, 3. The case of n∗ > 3 follows by using the same techniques as in the case of n∗ = 3 and is omitted because of the complexity of the control law that increases with n∗ . In Section 6.5 we consider the design and analysis of a wide class of direct

6.2. SIMPLE DIRECT MRAC SCHEMES

315

MRAC schemes with normalized adaptive laws for plants with arbitrary but known relative degree. The design of indirect MRAC with unnormalized and normalized adaptive laws is considered in Section 6.6. In Section 6.7, we briefly summarize some efforts and alternative approaches to relax some of the basic assumptions used in MRAC that include the minimum phase, known relative degree and upper bound on the order of the plant. In Section 6.8, we present all the long and more complicated proofs of theorems and lemmas.

6.2

Simple Direct MRAC Schemes

In this section, we use several examples to illustrate the design and analysis of some simple direct MRAC schemes with unnormalized adaptive laws. We concentrate on the SPR-Lyapunov approach for designing the adaptive laws. This approach dominated the literature of adaptive control for continuoustime plants with relative degree n∗ = 1 because of the simplicity of design and stability analysis [48, 85, 172, 201].

6.2.1

Scalar Example: Adaptive Regulation

Consider the following scalar plant: x˙ = ax + u, x(0) = x0

(6.2.1)

where a is a constant but unknown. The control objective is to determine a bounded function u = f (t, x) such that the state x(t) is bounded and converges to zero as t → ∞ for any given initial condition x0 . Let −am be the desired closed-loop pole where am > 0 is chosen by the designer. Control Law If the plant parameter a is known, the control law u = −k ∗ x

(6.2.2)

with k ∗ = a + am could be used to meet the control objective, i.e., with (6.2.2), the closed-loop plant is x˙ = −am x

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

whose equilibrium xe = 0 is e.s. in the large. Because a is unknown, k ∗ cannot be calculated and, therefore, (6.2.2) cannot be implemented. A possible procedure to follow in the unknown parameter case is to use the same control law as given in (6.2.2) but with k ∗ replaced by its estimate k(t), i.e., we use u = −k(t)x

(6.2.3)

and search for an adaptive law to update k(t) continuously with time. Adaptive Law The adaptive law for generating k(t) is developed by viewing the problem as an on-line identification problem for k ∗ . This is accomplished by first obtaining an appropriate parameterization for the plant (6.2.1) in terms of the unknown k ∗ and then using a similar approach as in Chapter 4 to estimate k ∗ on-line. We illustrate this procedure below. We add and subtract the desired control input −k ∗ x in the plant equation to obtain x˙ = ax − k ∗ x + k ∗ x + u. Because a − k ∗ = −am we have x˙ = −am x + k ∗ x + u or

1 (u + k ∗ x) (6.2.4) s + am Equation (6.2.4) is a parameterization of the plant equation (6.2.1) in terms of the unknown controller parameter k ∗ . Because x, u are measured and am > 0 is known, a wide class of adaptive laws may be generated by simply using Tables 4.1 to 4.3 of Chapter 4. It turns out that the adaptive laws developed for (6.2.4) using the SPR-Lyapunov design approach without normalization simplify the stability analysis of the resulting closed-loop adaptive control scheme considerably. Therefore, as a starting point, we concentrate on the simple case and deal with the more general case that involves a wide class of adaptive laws in later sections. 1 is SPR we can proceed with the SPR-Lyapunov design Because s+a m approach of Chapter 4 and generate the estimate x ˆ of x as x=

x ˆ=

1 1 [kx + u] = (0) s + am s + am

(6.2.5)

6.2. SIMPLE DIRECT MRAC SCHEMES

317

where the last equality is obtained by substituting the control law u = −kx. If we now choose x ˆ(0) = 0, we have x ˆ(t) ≡ 0, ∀t ≥ 0, which implies that the estimation error ²1 defined as ²1 = x − x ˆ is equal to the regulation error, i.e., ²1 = x, so that (6.2.5) does not have to be implemented to generate x ˆ. Substituting for the control u = −k(t)x in (6.2.4), we obtain the error equation that relates the parameter error k˜ = k − k ∗ with the estimation error ²1 = x, i.e., ˜ ²˙1 = −am ²1 − kx, ²1 = x (6.2.6) or ²1 =

³ ´ 1 ˜ −kx s + am

As demonstrated in Chapter 4, the error equation (6.2.6) is in a convenient form for choosing an appropriate Lyapunov function to design the adaptive law for k(t). We assume that the adaptive law is of the form ˙ k˜ = k˙ = f1 (²1 , x, u)

(6.2.7)

where f1 is some function to be selected, and propose ³ ´ k˜2 ²2 V ²1 , k˜ = 1 + 2 2γ

(6.2.8)

for some γ > 0 as a potential Lyapunov function for the system (6.2.6), (6.2.7). The time derivative of V along the trajectory of (6.2.6), (6.2.7) is given by ˜ ˜ 1 x + kf1 V˙ = −am ²21 − k² (6.2.9) γ Choosing f1 = γ²1 x, i.e., k˙ = γ²1 x = γx2 ,

k(0) = k0

(6.2.10)

we have V˙ = −am ²21 ≤ 0

(6.2.11)

Analysis Because V is a positive definite function and V˙ ≤ 0, we have V ∈ L∞ , which implies that ²1 , k˜ ∈ L∞ . Because ²1 = x, we also have that x ∈ L∞ and therefore all signals in the closed-loop plant are bounded.

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

Furthermore, ²1 = x ∈ L2 and ²˙1 = x˙ ∈ L∞ (which follows from (6.2.6) ) imply, according to Lemma 3.2.5, that ²1 (t) = x(t) → 0 as t → ∞. From ˙ x(t) → 0 and the boundedness of k, we establish that k(t) → 0, u(t) → 0 as t → ∞. We have shown that the combination of the control law (6.2.3) with the adaptive law (6.2.10) meets the control objective in the sense that it forces the plant state to converge to zero while guaranteeing signal boundedness. It is worth mentioning that as in the simple parameter identification examples considered in Chapter 4, we cannot establish that k(t) converges to k ∗ , i.e., that the pole of the closed-loop plant converges to the desired one given by −am . The lack of parameter convergence is less crucial in adaptive control than in parameter identification because in most cases, the control objective can be achieved without requiring the parameters to converge to their true values. The simplicity of this scalar example allows us to solve for ²1 = x explicitly, and study the properties of k(t), x(t) as they evolve with time. We can verify that 2ce−ct ²1 (0), ²1 = x c + k0 − a + (c − k0 + a) e−2ct £ ¤ c (c + k0 − a) e2ct − (c − k0 + a) k(t) = a + (c + k0 − a) e2ct + (c − k0 + a)

²1 (t) =

(6.2.12)

where c2 = γx20 + (k0 − a)2 , satisfy the differential equations (6.2.6) and (6.2.10) of the closed-loop plant. Equation (6.2.12) can be used to investigate the effects of initial conditions and adaptive gain γ on the transient and asymptotic behavior of x(t), k(t). We have limt→∞ k(t) = a + c if c > 0 and limt→∞ k(t) = a − c if c < 0, i.e., q

lim k(t) = k∞ = a +

t→∞

γx20 + (k0 − a)2

(6.2.13)

Therefore, for x0 6= 0, k(t) converges to a stabilizing gain whose value depends on γ and the initial condition x0 , k0 . It is clear from (6.2.13) that the value of k∞ is independent of whether k0 is a destabilizing gain, i.e., 0 < k0 < a, or a stabilizing one, i.e., k0 > a, as long as (k0 − a)2 is the same. The use of different k0 , however, will affect the transient behavior as it is obvious from (6.2.12). In the limit as t → ∞, the closed-loop pole converges

6.2. SIMPLE DIRECT MRAC SCHEMES

319

r = 0 - lu- 1 + Σ s−a − 6

x

µ ¡ ¡ ¾ k(t) ¡¡

¡ k

1 ¾ s 6

-

?

(•)2 γ

¾

k(0) Figure 6.2 Block diagram for implementing the adaptive controller (6.2.14). to − (k∞ − a), which may be different from −am . Because the control objective is to achieve signal boundedness and regulation of the state x(t) to zero, the convergence of k(t) to k ∗ is not crucial. Implementation The adaptive control scheme developed and analyzed above is given by the following equations: u = −k(t)x,

k˙ = γ²1 x = γx2 ,

k(0) = k0

(6.2.14)

where x is the measured state of the plant. A block diagram for implementing (6.2.14) is shown in Figure 6.2. The design parameters in (6.2.14) are the initial parameter k0 and the adaptive gain γ > 0. For signal boundedness and asymptotic regulation of x to zero, our analysis allows k0 , γ to be arbitrary. It is clear, however, from (6.2.12) that their values affect the transient performance of the closed-loop plant as well as the steady-state value of the closed-loop pole. For a given k0 , x 0 = 6 0, large γ leads to a larger value of c in (6.2.12) and, therefore, to a faster convergence of x(t) to zero. Large γ, however, may make the differential equation for k “stiff” (i.e., k˙ large) that will require a very small step size or sampling period to implement it on a digital computer. Small sampling periods make the adaptive scheme more sensitive to measurement noise and modeling errors.

320

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

Remark 6.2.1 In the proceeding example, we have not used any reference model to describe the desired properties of the closed-loop system. A reasonable choice for the reference model would be x˙ m = −am xm ,

xm (0) = xm0

(6.2.15)

which, by following exactly the same procedure, would lead to the adaptive control scheme u = −k(t)x,

k˙ = γe1 x

where e1 = x − xm . If xm0 6= x0 , the use of (6.2.15) will affect the transient behavior of the tracking error but will have no effect on the asymptotic properties of the closed-loop scheme because xm converges to zero exponentially fast.

6.2.2

Scalar Example: Adaptive Tracking

Consider the following first order plant: x˙ = ax + bu

(6.2.16)

where a, b are unknown parameters but the sign of b is known. The control objective is to choose an appropriate control law u such that all signals in the closed-loop plant are bounded and x tracks the state xm of the reference model given by x˙ m = −am xm + bm r i.e., xm =

bm r s + am

(6.2.17)

for any bounded piecewise continuous signal r(t), where am > 0, bm are known and xm (t), r(t) are measured at each time t. It is assumed that am , bm and r are chosen so that xm represents the desired state response of the plant. Control Law For x to track xm for any reference input signal r(t), the control law should be chosen so that the closed-loop plant transfer function

6.2. SIMPLE DIRECT MRAC SCHEMES

321

from the input r to output x is equal to that of the reference model. We propose the control law u = −k ∗ x + l∗ r (6.2.18) where k ∗ , l∗ are calculated so that x(s) bl∗ bm xm (s) = = = r(s) s − a + bk ∗ s + am r(s)

(6.2.19)

Equation (6.2.19) is satisfied if we choose bm am + a , k∗ = (6.2.20) b b provided of course that b 6= 0, i.e., the plant (6.2.16) is controllable. The control law (6.2.18), (6.2.20) guarantees that the transfer function of the closed-loop plant, i.e., x(s) r(s) is equal to that of the reference model. Such a transfer function matching guarantees that x(t) = xm (t), ∀t ≥ 0 when x(0) = xm (0) or |x(t) − xm (t)| → 0 exponentially fast when x(0) 6= xm (0), for any bounded reference signal r(t). When the plant parameters a, b are unknown, (6.2.18) cannot be implemented. Therefore, instead of (6.2.18), we propose the control law l∗ =

u = −k(t)x + l(t)r

(6.2.21)

where k(t), l(t) is the estimate of k ∗ , l∗ , respectively, at time t, and search for an adaptive law to generate k(t), l(t) on-line. Adaptive Law As in Example 6.2.1, we can view the problem as an online identification problem of the unknown constants k ∗ , l∗ . We start with the plant equation (6.2.16) which we express in terms of k ∗ , l∗ by adding and subtracting the desired input term −bk ∗ x + bl∗ r to obtain x˙ = −am x + bm r + b (k ∗ x − l∗ r + u) i.e., x= Because xm =

bm s+am r

bm b r+ (k ∗ x − l∗ r + u) s + am s + am

(6.2.22)

is a known bounded signal, we express (6.2.22) in terms 4

of the tracking error defined as e = x − xm , i.e., e=

b (k ∗ x − l∗ r + u) s + am

(6.2.23)

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

Because b is unknown, equation (6.2.23) is in the form of the bilinear parametric model considered in Chapter 4, and may be used to choose an adaptive law directly from Table 4.4 of Chapter 4. Following the procedure of Chapter 4, the estimate eˆ of e is generated as eˆ =

1 ˆ 1 b (kx − lr + u) = (0) s + am s + am

(6.2.24)

where the last identity is obtained by substituting for the control law u = −k(t)x + l(t)r 4

Equation (6.2.24) implies that the estimation error, defined as ²1 = e − eˆ, can be simply taken to be the tracking error, i.e., ²1 = e, and, therefore, there is no need to generate eˆ. Furthermore, since eˆ is not generated, the estimate ˆb of b is not required. Substituting u = −k(t)x + l(t)r in (6.2.23) and defining the parameter 4 4 errors k˜ = k − k ∗ , ˜l = l − l∗ , we have ²1 = e = or

³

³ ´ b ˜ + ˜lr −kx s + am ´

˜ + ˜lr , ²˙1 = −am ²1 + b −kx

²1 = e = x − xm

(6.2.25)

As shown in Chapter 4, the development of the differential equation (6.2.25) relating the estimation error with the parameter error is a significant step in deriving the adaptive laws for updating k(t), l(t). We assume that the structure of the adaptive law is given by k˙ = f1 (²1 , x, r, u) ,

l˙ = f2 (²1 , x, r, u)

(6.2.26)

where the functions f1 , f2 are to be designed. As shown in Example 6.2.1, however, the use of the SPR-Lyapunov approach without normalization allows us to design an adaptive law for k, l and analyze the stability properties of the closed-loop system using a single Lyapunov function. For this reason, we proceed with the SPR-Lyapunov approach without normalization and postpone the use of other approaches that are based on the use of the normalized estimation error for later sections.

6.2. SIMPLE DIRECT MRAC SCHEMES

323

Consider the function 2 ³ ´ ˜2 ˜2 ˜ ˜l = ²1 + k |b| + l |b| V ²1 , k, 2 2γ1 2γ2

(6.2.27)

where γ1 , γ2 > 0 as a Lyapunov candidate for the system (6.2.25), (6.2.26). The time derivative V˙ along any trajectory of (6.2.25), (6.2.26) is given by ˜ ˜ ˜ 1 x + b˜l²1 r + |b|k f1 + |b|l f2 V˙ = −am ²21 − bk² γ1 γ2

(6.2.28)

Because |b| = bsgn(b), the indefinite terms in (6.2.28) disappear if we choose f1 = γ1 ²1 xsgn(b), f2 = −γ2 ²1 r sgn(b). Therefore, for the adaptive law k˙ = γ1 ²1 x sgn(b), l˙ = −γ2 ²1 r sgn(b)

(6.2.29)

V˙ = −am ²21

(6.2.30)

we have

Analysis Treating xm (t), r(t) in (6.2.25) as bounded functions of time, it follows from (6.2.27), (6.2.30) that V is a Lyapunov function for the thirdorder differential equation (6.2.25) and (6.2.29) where xm is treated as a bounded function of time, and the equilibrium ²1e = ee = 0, k˜e = 0, ˜le = 0 ˜ ˜l ∈ L∞ and ²1 ∈ L2 . Because ²1 = e = x − xm is u.s. Furthermore, ²1 , k, and xm ∈ L∞ , we also have x ∈ L∞ and u ∈ L∞ ; therefore, all signals in the closed-loop are bounded. Now from (6.2.25) we have ²˙1 ∈ L∞ , which, together with ²1 ∈ L2 , implies that ²1 (t) = e(t) → 0 as t → ∞. We have established that the control law (6.2.21) together with the adaptive law (6.2.29) guarantee boundedness for all signals in the closed-loop system. In addition, the plant state x(t) tracks the state of the reference model xm asymptotically with time for any reference input signal r, which is bounded and piecewise continuous. These results do not imply that k(t) → k ∗ and l(t) → l∗ as t → ∞, i.e., the transfer function of the closed-loop plant may not approach that of the reference model as t → ∞. To achieve such a result, the reference input r has to be sufficiently rich of order 2. For example, r(t) = sin ωt for some ω 6= 0 guarantees the exponential convergence of x(t) to xm (t) and of k(t), l(t) to k ∗ , l∗ , respectively. In general, a sufficiently rich reference input r(t) is not desirable especially

324

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL Reference Model

r

7 ¶ - l(t) ··

+ Σl

l(t) 1 ¾ l(0) s 6

−γ2 sgn(b) 6

xm

bm s + am

-

u-

− 6

b s−a

? − Σl + x 6

e = ²1-

¶ 7

k(t) ¾ k(t) ·· 1 ¾ s

k(0)

- × ¾

6

γ1 sgn(b) 6

- ×

Figure 6.3 Block diagram for implementing the adaptive law (6.2.21) and (6.2.29). in cases where the control objective involves tracking of signals that are not rich in frequencies. Parameter convergence and the conditions the reference input r has to satisfy are discussed later on in this chapter. Implementation The MRAC control law (6.2.21), (6.2.29) can be implemented as shown in Figure 6.3. The adaptive gains γ1 , γ2 are designed by following similar considerations as in the previous examples. The initial conditions l(0), k(0) are chosen to be any a priori guess of the unknown parameters l∗ , k ∗ , respectively. Small initial parameter error usually leads to better transient behavior. As we mentioned before, the reference model and input r are designed so that xm describes the desired trajectory to be followed by the plant state. Remark 6.2.2 The assumption that the sign of b is known may be relaxed by using the techniques of Chapter 4 summarized in Table 4.4 and is left as an exercise for the reader (see Problem 6.3).

6.2. SIMPLE DIRECT MRAC SCHEMES

6.2.3

325

Vector Case: Full-State Measurement

Let us now consider the nth order plant x˙ = Ax + Bu,

x ∈ Rn

(6.2.31)

where A ∈ Rn×n , B ∈ Rn×q are unknown constant matrices and (A, B) is controllable. The control objective is to choose the input vector u ∈ Rq such that all signals in the closed-loop plant are bounded and the plant state x follows the state xm ∈ Rn of a reference model specified by the LTI system x˙ m = Am xm + Bm r

(6.2.32)

where Am ∈ Rn×n is a stable matrix, Bm ∈ Rn×q , and r ∈ Rq is a bounded reference input vector. The reference model and input r are chosen so that xm (t) represents a desired trajectory that x has to follow. Control Law If the matrices A, B were known, we could apply the control law u = −K ∗ x + L∗ r (6.2.33) and obtain the closed-loop plant x˙ = (A − BK ∗ )x + BL∗ r

(6.2.34)

Hence, if K ∗ ∈ Rq×n and L∗ ∈ Rq×q are chosen to satisfy the algebraic equations A − BK ∗ = Am , BL∗ = Bm (6.2.35) then the transfer matrix of the closed-loop plant is the same as that of the reference model and x(t) → xm (t) exponentially fast for any bounded reference input signal r(t). We should note that given the matrices A, B, Am , Bm , no K ∗ , L∗ may exist to satisfy the matching condition (6.2.35) indicating that the control law (6.2.33) may not have enough structural flexibility to meet the control objective. In some cases, if the structure of A, B is known, Am , Bm may be designed so that (6.2.35) has a solution for K ∗ , L∗ . Let us assume that K ∗ , L∗ in (6.2.35) exist, i.e., that there is sufficient structural flexibility to meet the control objective, and propose the control law u = −K(t)x + L(t)r (6.2.36)

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

where K(t), L(t) are the estimates of K ∗ , L∗ , respectively, to be generated by an appropriate adaptive law. Adaptive Law By adding and subtracting the desired input term, namely, −B(K ∗ x−L∗ r) in the plant equation and using (6.2.35), we obtain x˙ = Am x + Bm r + B(K ∗ x − L∗ r + u)

(6.2.37)

which is the extension of the scalar equation (6.2.22) in Example 6.2.2 to the vector case. Following the same procedure as in Section 6.2.2, we can 4 ˜ = show that the tracking error e = x − xm and parameter error K K − K ∗, 4 ˜ = L − L∗ satisfy the equation L ˜ + Lr) ˜ e˙ = Am e + B(−Kx

(6.2.38)

which also depends on the unknown matrix B. In the scalar case we manage to get away with the unknown B by assuming that its sign is known. An extension of the scalar assumption of Section 6.2.2 to the vector case is as follows: Let us assume that L∗ is either positive definite or negative definite and Γ−1 = L∗ sgn(l), where l = 1 if L∗ is positive definite and l = −1 if L∗ is negative definite. Then B = Bm L∗−1 and (6.2.38) becomes ˜ + Lr) ˜ e˙ = Am e + Bm L∗−1 (−Kx We propose the following Lyapunov function candidate ˜ L) ˜ = e> P e + tr[K ˜ > ΓK ˜ +L ˜ > ΓL] ˜ V (e, K, where P = P > > 0 satisfies the Lyapunov equation P Am + A> m P = −Q for some Q = Q> > 0. Then, ˜ + Lr) ˜ + 2tr[K ˜ > ΓK ˜˙ + L ˜ > ΓL] ˜˙ V˙ = −e> Qe + 2e> P Bm L∗−1 (−Kx Now > > ˜ = tr[x> K ˜ > ΓBm ˜ > ΓBm e> P Bm L∗−1 Kx P e]sgn(l) = tr[K P ex> ]sgn(l)

6.2. SIMPLE DIRECT MRAC SCHEMES

327

and ˜ = tr[L ˜ > ΓB > P er> ]sgn(l) e> P Bm L∗−1 Lr m Therefore, for > > ˜˙ = K˙ = Bm ˜˙ = L˙ = −Bm K P ex> sgn(l), L P er> sgn(l)

(6.2.39)

we have V˙ = −e> Qe Analysis From the properties of V, V˙ , we establish as in the scalar case that K(t), L(t), e(t) are bounded and that e(t) → 0 as t → ∞. Implementation The adaptive control scheme developed is given by (6.2.36) and (6.2.39). The matrix Bm P acts as an adaptive gain matrix, where P is obtained by solving the Lyapunov equation P Am + A> m P = −Q for some arbitrary Q = Q> > 0. Different choices of Q will not affect boundedness and the asymptotic behavior of the scheme, but they will affect the transient response. The assumption that the unknown L∗ in the matching equation BL∗ = Bm is either positive or negative definite imposes an additional restriction on the structure and elements of B, Bm . Because B is unknown this assumption may not be realistic in some applications. The case where B is completely unknown is treated in [172] using the adaptive control law u = −L(t)K(t)x + L(t)r > > K˙ = Bm P ex> , L˙ = −LBm eu> L

(6.2.40)

The result established, however, is only local which indicates that for stability K(0), L(0) have to be chosen close to the equilibrium Ke = K ∗ , Le = L∗ of (6.2.40). Furthermore, K ∗ , L∗ are required to satisfy the matching equations A − BL∗ K ∗ = Am , BL∗ = Bm .

6.2.4

Nonlinear Plant

The procedure of Sections 6.2.1 to 6.2.3 can be extended to some special classes of nonlinear plants as demonstrated briefly by using the following nonlinear example x˙ = af (x) + bg(x)u (6.2.41)

328

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

where a, b are unknown scalars, f (x), g(x) are known functions with g(x) > c > 0 ∀x ∈ R1 and some constant c > 0. The sgn(b) is known and f (x) is bounded for bounded x. It is desired that x tracks the state xm of the reference model given by x˙ m = −am xm + bm r for any bounded reference input signal r. Control Law

If a, b were known, the control law u=

with

1 [k ∗ f (x) + k2∗ x + l∗ r] g(x) 1

(6.2.42)

am bm a k1∗ = − , k2∗ = − , l∗ = b b b

could meet the control objective exactly. For the case of a, b unknown, we propose a control law of the same form as (6.2.42) but with adjustable gains, i.e., we use 1 u= [k1 (t)f (x) + k2 (t)x + l(t)r] (6.2.43) g(x) where k1 , k2 , l, are the estimates of the unknown controller gains k1∗ , k2∗ , l∗ respectively to be generated by an adaptive law. Adaptive Law As in the previous examples, we first rewrite the plant equation in terms of the unknown controller gains k1∗ , k2∗ , l∗ , i.e., substituting for a = −bk1∗ and adding and subtracting the term b (k1∗ f (x) + k2∗ x + l∗ r) in (6.2.41) and using the equation bl∗ = bm , bk2∗ = −am , we obtain x˙ = −am x + bm r + b [−k1∗ f (x) − k2∗ x − l∗ r + g(x)u] 4 4 4 4 If we let e = x − xm , k˜1 = k1 − k1∗ , k˜2 = k2 − k2∗ , ˜l = l − l∗ to be the tracking and parameter errors, we can show as before that the tracking error satisfies the differential equation

³

´

e˙ = −am e + b k˜1 f (x) + k˜2 x + ˜lr , ²1 = e

6.2. SIMPLE DIRECT MRAC SCHEMES

329

which we can use as in Section 6.2.2 to develop the adaptive laws k˙ 1 = −γ1 ef (x) sgn(b) k˙ 2 = −γ2 ex sgn(b) l˙ = −γ3 er sgn(b)

(6.2.44)

where γi > 0, i = 1, 2, 3 are the adaptive gains. Analysis We can establish that all signals in the closed-loop plant (6.2.41), (6.2.43), and (6.2.44) are bounded and that |e(t)| = |x(t) − xm (t)| → 0 as t → ∞ by using the Lyapunov function ˜l2 e2 k˜2 k˜2 V (e, k˜1 , k˜2 , ˜l) = + 1 |b| + 2 |b| + |b| 2 2γ1 2γ2 2γ3 in a similar way as in Section 6.2.2. The choice of the control law to cancel the nonlinearities and force the plant to behave as an LTI system is quite obvious for the case of the plant (6.2.41). Similar techniques may be used to deal with some more complicated nonlinear problems where the choice of the control law in the known and unknown parameter case is less obvious [105]. Remark 6.2.3 The simple adaptive control schemes presented in this section have the following characteristics: (i) The adaptive laws are developed using the SPR-Lyapunov design approach and are driven by the estimation error rather than the normalized estimation error. The estimation error is equal to the regulation or tracking error that is to be driven to zero as a part of the control objective. (ii) The design of the adaptive law and the stability analysis of the closedloop adaptive scheme is accomplished by using a single Lyapunov function. (iii) The full state vector is available for measurement.

Another approach is to use the procedure of Chapter 4 and develop adaptive laws based on the SPR-Lyapunov method that are driven by normalized

330

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

estimation errors. Such schemes, however, are not as easy to analyze as the schemes with unnormalized adaptive laws developed in this section. The reason is that the normalized estimation error is not simply related to the regulation or tracking error and additional stability arguments are needed to complete the analysis of the respective adaptive control scheme. The distinction between adaptive schemes with normalized and unnormalized adaptive laws is made clear in this chapter by analyzing them in separate sections. Some of the advantages and disadvantages of normalized and unnormalized adaptive laws are discussed in the sections to follow. The assumption of full state measurement in the above examples is relaxed in the following sections where we formulate and solve the general MRAC problem.

6.3

MRC for SISO Plants

In Section 6.2 we used several examples to illustrate the design and analysis of MRAC schemes for plants whose state vector is available for measurement. The design of MRAC schemes for plants whose output rather than the full state is available for measurement follows a similar procedure as that used in Section 6.2. This design procedure is based on combining a control law whose form is the same as the one we would use in the known parameter case with an adaptive law that provides on-line estimates for the controller parameters. In the general case, the design of the control law is not as straightforward as it appears to be in the case of the examples of Section 6.2. Because of this reason, we use this section to formulate the MRC problem for a general class of LTI SISO plants and solve it for the case where the plant parameters are known exactly. The significance of the existence of a control law that solves the MRC problem is twofold: First it demonstrates that given a set of assumptions about the plant and reference model, there is enough structural flexibility to meet the control objective; second, it provides the form of the control law that is to be combined with an adaptive law to form MRAC schemes in the case of unknown plant parameters to be treated in the sections to follow.

6.3. MRC FOR SISO PLANTS

6.3.1

331

Problem Statement

Consider the SISO, LTI plant described by the vector differential equation x˙ p = Ap xp + Bp up , yp =

xp (0) = x0

Cp> xp

(6.3.1)

where xp ∈ Rn ; yp , up ∈ R1 and Ap , Bp , Cp have the appropriate dimensions. The transfer function of the plant is given by yp = Gp (s)up

(6.3.2)

with Gp (s) expressed in the form Gp (s) = kp

Zp (s) Rp (s)

(6.3.3)

where Zp , Rp are monic polynomials and kp is a constant referred to as the high frequency gain. The reference model, selected by the designer to describe the desired characteristics of the plant, is described by the differential equation x˙ m = Am xm + Bm r, ym =

xm (0) = xm0

> Cm xm

(6.3.4)

where xm ∈ Rpm for some integer pm ; ym , r ∈ R1 and r is the reference input which is assumed to be a uniformly bounded and piecewise continuous function of time. The transfer function of the reference model given by ym = Wm (s)r is expressed in the same form as (6.3.3), i.e., Wm (s) = km

Zm (s) Rm (s)

(6.3.5)

where Zm (s), Rm (s) are monic polynomials and km is a constant. The MRC objective is to determine the plant input up so that all signals are bounded and the plant output yp tracks the reference model output ym as close as possible for any given reference input r(t) of the class defined

332

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

above. We refer to the problem of finding the desired up to meet the control objective as the MRC problem. In order to meet the MRC objective with a control law that is implementable, i.e., a control law that is free of differentiators and uses only measurable signals, we assume that the plant and reference model satisfy the following assumptions: Plant Assumptions P1. Zp (s) is a monic Hurwitz polynomial of degree mp P2. An upper bound n of the degree np of Rp (s) P3. the relative degree n∗ = np − mp of Gp (s), and P4. the sign of the high frequency gain kp are known Reference Model Assumptions: M1. Zm (s), Rm (s) are monic Hurwitz polynomials of degree qm , pm , respectively, where pm ≤ n. M2. The relative degree n∗m = pm − qm of Wm (s) is the same as that of Gp (s), i.e., n∗m = n∗ . Remark 6.3.1 Assumption P1 requires the plant transfer function Gp (s) to be minimum phase. We make no assumptions, however, about the location of the poles of Gp (s), i.e., the plant is allowed to have unstable poles. We allow the plant to be uncontrollable or unobservable, i.e., we allow common zeros and poles in the plant transfer function. Because, by assumption P1, all the plant zeros are in C − , any zero-pole cancellation can only occur in C − , which implies that the plant (6.3.1) is both stabilizable and detectable. The minimum phase assumption (P1) is a consequence of the control objective which is met by designing an MRC control law that cancels the zeros of the plant and replaces them with those of the reference model in an effort to force the closed-loop plant transfer function from r to yp to be equal to Wm (s). For stability, such cancellations should occur in C − which implies that Zp (s) should satisfy assumption P1. As we will show in Section 6.7, assumptions P3, P4 can be relaxed at the expense of more complex control laws.

6.3. MRC FOR SISO PLANTS

6.3.2

333

MRC Schemes: Known Plant Parameters

In addition to assumptions P1 to P4 and M1, M2, let us also assume that the plant parameters, i.e., the coefficients of Gp (s) are known exactly. Because the plant is LTI and known, the design of the MRC scheme is achieved using linear system theory. The MRC objective is met if up is chosen so that the closed-loop transfer function from r to yp has stable poles and is equal to Wm (s), the transfer function of the reference model. Such a transfer function matching guarantees that for any reference input signal r(t), the plant output yp converges to ym exponentially fast. A trivial choice for up is the cascade open-loop control law up = C(s)r, C(s) =

km Zm (s) Rp (s) kp Rm (s) Zp (s)

(6.3.6)

which leads to the closed-loop transfer function yp km Zm Rp kp Zp = = Wm (s) r kp Rm Zp Rp

(6.3.7)

This control law, however, is feasible only when Rp (s) is Hurwitz. Otherwise, (6.3.7) may involve zero-pole cancellations outside C − , which will lead to unbounded internal states associated with non-zero initial conditions [95]. In addition, (6.3.6) suffers from the usual drawbacks of open loop control such as deterioration of performance due to small parameter changes and inexact zero-pole cancellations. Instead of (6.3.6), let us consider the feedback control law up = θ1∗>

α(s) α(s) up + θ2∗> yp + θ3∗ yp + c∗0 r Λ(s) Λ(s)

(6.3.8)

shown in Figure 6.4 where 4

£

α(s) = αn−2 (s) = sn−2 , sn−3 , . . . , s, 1 4

α(s) = 0

¤>

for n ≥ 2 for n = 1

c∗0 , θ3∗ ∈ R1 ; θ1∗ , θ2∗ ∈ Rn−1 are constant parameters to be designed and Λ(s) is an arbitrary monic Hurwitz polynomial of degree n−1 that contains Zm (s) as a factor, i.e., Λ(s) = Λ0 (s)Zm (s)

334

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL r- c∗ 0

+ - Σn +£± AK+ 6A £+

£

£

£

£

£

£

£

up θ1∗>

A

yp -

- Gp (s)

α(s) ¾ Λ(s) θ2∗>

α(s) ¾ Λ(s) θ3∗ ¾

Figure 6.4 Structure of the MRC scheme (6.3.8). which implies that Λ0 (s) is monic, Hurwitz and of degree n0 = n − 1 − qm . The controller parameter vector h

θ∗ = θ1∗> , θ2∗> , θ3∗ , c∗0

i>

∈ R2n

is to be chosen so that the transfer function from r to yp is equal to Wm (s). The I/O properties of the closed-loop plant shown in Figure 6.4 are described by the transfer function equation yp = Gc (s)r

(6.3.9)

where ∗

Gc (s) =

Λ

£¡

2

c kp Z p Λ ¢ 0 ¡ ¢¤ ∗> Λ − θ1 α(s) Rp − kp Zp θ2∗> α(s) + θ3∗ Λ

(6.3.10)

We can now meet the control objective if we select the controller parameters θ1∗ , θ2∗ , θ3∗ , c∗0 so that the closed-loop poles are stable and the closed-loop transfer function Gc (s) = Wm (s), i.e., ∗

Λ

2

Zm c kp Z p Λ ¢ 0 ¡ ∗> ¢¤ = km ∗> ∗ Rm Λ − θ1 α Rp − kp Zp θ2 α + θ3 Λ

£¡

(6.3.11)

is satisfied for all s ∈ C. Because the degree of the denominator of Gc (s) is np + 2n − 2 and that of Rm (s) is pm ≤ n, for the matching equation (6.3.11) to hold, an additional np + 2n − 2 − pm zero-pole cancellations must occur in Gc (s). Now because Zp (s) is Hurwitz by assumption and Λ(s) = Λ0 (s)Zm (s)

6.3. MRC FOR SISO PLANTS

335

is designed to be Hurwitz, it follows that all the zeros of Gc (s) are stable and therefore any zero-pole cancellation can only occur in C − . Choosing c∗0 =

km kp

(6.3.12)

and using Λ(s) = Λ0 (s)Zm (s) the matching equation (6.3.11) becomes ³

³

´

´

Λ − θ1∗> α Rp − kp Zp θ2∗> α + θ3∗ Λ = Zp Λ0 Rm

(6.3.13)

or ³

´

θ1∗>α(s)Rp (s) + kp θ2∗>α(s) + θ3∗Λ(s) Zp (s) = Λ(s)Rp (s) − Zp (s)Λ0 (s)Rm (s) (6.3.14) Equating the coefficients of the powers of s on both sides of (6.3.14), we can express (6.3.14) in terms of the algebraic equation S θ¯∗ = p h

(6.3.15)

i>

where θ¯∗ = θ1∗> , θ2∗> , θ3∗ , S is an (n + np − 1) × (2n − 1) matrix that depends on the coefficients of Rp , kp Zp and Λ, and p is an n + np − 1 vector with the coefficients of ΛRp −Zp Λ0 Rm . The existence of θ¯∗ to satisfy (6.3.15) and, therefore, (6.3.14) will very much depend on the properties of the matrix S. For example, if n > np , more than one θ¯∗ will satisfy (6.3.15), whereas if n = np and S is nonsingular, (6.3.15) will have only one solution.

Remark 6.3.2 For the design of the control input (6.3.8), we assume that n ≥ np . Because the plant is known exactly, there is no need to assume an upper bound for the degree of the plant, i.e., because np is known n can be set equal to np . We use n ≥ np on purpose in order to use the result in the unknown plant parameter case treated in Sections 6.4 and 6.5, where only the upper bound n for np is known. Remark 6.3.3 Instead of using (6.3.15), one can solve (6.3.13) for θ1∗ , θ2∗ , θ3∗ as follows: Dividing both sides of (6.3.13) by Rp (s), we obtain Ã

Λ−

θ1∗> α

∆∗ Zp − kp (θ2∗> α + θ3∗ Λ) = Zp Q + kp Rp Rp

!

336

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL where Q(s) (of degree n − 1 − mp ) is the quotient and kp ∆∗ (of degree at most np − 1) is the remainder of Λ0 Rm /Rp , respectively. Then the solution for θi∗ , i = 1, 2, 3 can be found by inspection, i.e., θ1∗> α(s) = Λ(s) − Zp (s)Q(s) θ2∗> α(s) + θ3∗ Λ(s) =

Q(s)Rp (s) − Λ0 (s)Rm (s) kp

(6.3.16) (6.3.17)

where the equality in the second equation is obtained by substituting k ∆∗ for ∆∗ (s) using the identity Λ0RRpm = Q + pRp . The parameters θi∗ , i = 1, 2, 3 can now be obtained directly by equating the coefficients of the powers of s on both sides of (6.3.16), (6.3.17). Equations (6.3.16) and (6.3.17) indicate that in general the controller parameters θi∗ , i = 1, 2, 3 are nonlinear functions of the coefficients of the plant polynomials Zp (s), Rp (s) due to the dependence of Q(s) on the coefficients of Rp (s). When n = np and n∗ = 1, however, Q(s) = 1 and the θi∗ ’s are linear functions of the coefficients of Zp (s), Rp (s). Lemma 6.3.1 Let the degrees of Rp , Zp , Λ, Λ0 and Rm be as specified in (6.3.8). Then (i) The solution θ¯∗ of (6.3.14) or (6.3.15) always exists. (ii) In addition if Rp , Zp are coprime and n = np , then the solution θ¯∗ is unique. ¯ p (s)h(s) and Zp (s) = Z¯p (s)h(s) and R ¯ p (s), Z¯p (s) be coprime, Proof Let Rp = R where h(s) is a monic polynomial of degree r0 (with 0 ≤ r0 ≤ mp ). Because Zp (s) is Hurwitz, it follows that h(s) is also Hurwitz. If Rp , Zp are coprime, h(s) = 1, i.e., r0 = 0. If Rp , Zp are not coprime, r0 ≥ 1 and h(s) is their common factor. We can now write (6.3.14) as ¯ p + kp (θ2∗> α + θ3∗ Λ)Z¯p = ΛR ¯ p − Z¯p Λ0 Rm θ1∗> αR

(6.3.18)

by canceling h(s) from both sides of (6.3.14). Because h(s) is Hurwitz, the cancellation occurs in C − . Equation (6.3.18) leads to np +n−r0 −2 algebraic equations with ¯ p − Z¯p Λ0 Rm is np +n−r0 −2 2n−1 unknowns. It can be shown that the degree of ΛR ¯ p , Z¯p are coprime, because of the cancellation of the term snp +n−r0 −1 . Because R it follows from Theorem 2.3.1 that there exists unique polynomials a0 (s), b0 (s) of degree n − 2, np − r0 − 1 respectively such that ¯ p (s) + b0 (s)Z¯p (s) = Λ(s)R ¯ p (s) − Z¯p (s)Λ0 (s)Rm (s) a0 (s)R

(6.3.19)

6.3. MRC FOR SISO PLANTS

337

is satisfied for n ≥ 2. It now follows by inspection that

and

θ1∗> α(s) = f (s)Z¯p (s) + a0 (s)

(6.3.20)

¯ p (s) + b0 (s) kp (θ2∗> α(s) + θ3∗ Λ(s)) = −f (s)R

(6.3.21)

satisfy (6.3.18), where f (s) is any given polynomial of degree nf = n − np + r0 − 1. Hence, the solution θ1∗ , θ2∗ , θ3∗ of (6.3.18) can be obtained as follows: We first solve (6.3.19) for a0 (s), b0 (s). We then choose an arbitrary polynomial f (s) of degree nf = n − np + r0 − 1 and calculate θ1∗ , θ2∗ , θ3∗ from (6.3.20), (6.3.21) by equating coefficients of the powers of s. Because f (s) is arbitrary, the solution θ1∗ , θ2∗ , θ3∗ is not unique. If, however, n = np and r0 = 0, i.e., Rp , Zp are coprime, then f (s) = 0 and θ1∗> α(s) = a0 (s), kp (θ2∗> α(s) + θ3∗ Λ(s)) = b0 (s) which implies that the solution θ1∗ , θ2∗ , θ3∗ is unique due to the uniqueness of a0 (s), b0 (s). If n = np = 1, then α(s) = 0, Λ(s) = 1, θ1∗ = θ2∗ = 0 and θ3∗ given by (6.3.18) is unique. 2

Remark 6.3.4 It is clear from (6.3.12), (6.3.13) that the control law (6.3.8) places the poles of the closed-loop plant at the roots of the polynomial Zp (s)Λ0 (s)Rm (s) and changes the high frequency gain from kp to km by using the feedforward gain c∗0 . Therefore, the MRC scheme can be viewed as a special case of a general pole placement scheme where the desired closed-loop characteristic equation is given by Zp (s)Λ0 (s)Rm (s) = 0 The transfer function matching (6.3.11) is achieved by canceling the zeros of the plant, i.e., Zp (s), and replacing them by those of the reference model, i.e., by designing Λ = Λ0 Zm . Such a cancellation is made possible by assuming that Zp (s) is Hurwitz and by designing Λ0 , Zm to have stable zeros. We have shown that the control law (6.3.8) guarantees that the closedloop transfer function Gc (s) of the plant from r to yp has all its poles in C − and in addition, Gc (s) = Wm (s). In our analysis we assumed zero initial conditions for the plant, reference model and filters. The transfer function matching, i.e., Gc (s) = Wm (s), together with zero initial conditions guarantee that yp (t) = ym (t), ∀t ≥ 0 and for any reference input r(t) that is

338

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

bounded and piecewise continuous. The assumption of zero initial conditions is common in most I/O control design approaches for LTI systems and is valid provided that any zero-pole cancellation in the closed-loop plant transfer function occurs in C − . Otherwise nonzero initial conditions may lead to unbounded internal states that correspond to zero-pole cancellations in C + . In our design we make sure that all cancellations in Gc (s) occur in C − by assuming stable zeros for the plant transfer function and by using stable filters in the control law. Nonzero initial conditions, however, will affect the transient response of yp (t). As a result we can no longer guarantee that yp (t) = ym (t) ∀t ≥ 0 but instead that yp (t) → ym (t) exponentially fast with a rate that depends on the closed-loop dynamics. We analyze the effect of initial conditions by using state space representations for the plant, reference model, and controller as follows: We begin with the following state-space realization of the control law (6.3.8): ω˙ 1 = F ω1 + gup ,

ω1 (0) = 0

ω˙ 2 = F ω2 + gyp ,

ω2 (0) = 0

up = θ

∗>

(6.3.22)

ω

where ω1 , ω2 ∈ Rn−1 , h

θ∗ = θ1∗> , θ2∗> , θ3∗ , c∗0     F =   

i>

, ω = [ω1> , ω2> , yp , r]> 

  −λn−2 −λn−3 −λn−4 . . . −λ0 1  1 0 0 ... 0      0  0 1 0 ... 0 , g =  .   .  .. .. ..   .. ..  .  . .  . . . 0 0 0 ... 1 0

(6.3.23)

λi are the coefficients of Λ(s) = sn−1 + λn−2 sn−2 + . . . + λ1 s + λ0 = det(sI − F ) α(s) α(s) and (F, g) is the state space realization of Λ(s) , i.e., (sI − F )−1 g = Λ(s) . The block diagram of the closed-loop plant with the control law (6.3.22) is shown in Figure 6.5.

6.3. MRC FOR SISO PLANTS r

339

yp + l up - Gp (s) Σ + + µ£± 6 ¡ ¡ + ? ? ¡ £ −1 ¡ £ (sI − F ) g (sI − F )−1 g

- c∗ 0

-

θ1∗> ¾ ω1 ω θ2∗> ¾ 2 θ3∗ ¾ Figure 6.5 Block diagram of the MRC scheme (6.3.22). We obtain the state-space representation of the overall closed-loop plant by augmenting the state xp of the plant (6.3.1) with the states ω1 , ω2 of the controller (6.3.22), i.e., Y˙ c = Ac Yc + Bc c∗0 r,

Yc (0) = Y0

yp = Cc> Yc where

(6.3.24)

h

> > Yc = x> p , ω1 , ω2

i>

∈ Rnp +2n−2







Ap + Bp θ3∗ Cp> Bp θ1∗> Bp θ2∗>   ∗ > ∗> gθ3 Cp F + gθ1 gθ2∗>  , Ac =  gCp> 0 F h



Bp   Bc =  g  0

(6.3.25)

i

Cc> = Cp> , 0, 0

and Y0 is the vector with initial conditions. We have already established that the transfer function from r to yp is given by yp (s) c∗0 kp Zp Λ2 ¢ ¡ ¢¤ = Wm (s) = £¡ r(s) Λ Λ − θ1∗> α Rp − kp Zp θ2∗> α + θ3∗ Λ which implies that Cc> (sI − Ac )−1 Bc c∗0 =

∗

£¡

Λ Λ−

θ1∗> α

2

c kp Zp Λ ¡ ¢¤ = Wm (s) ¢ 0 Rp − kp Zp θ2∗> α + θ3∗ Λ

340

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

and, therefore, h³

det (sI − Ac ) = Λ

´

³

´i

Λ − θ1∗> α Rp − kp Zp θ2∗> α + θ3∗ Λ

= ΛZp Λ0 Rm

where the last equality is obtained by using the matching equation (6.3.13). It is clear that the eigenvalues of Ac are equal to the roots of the polynomials Λ, Zp and Rm ; therefore, Ac is a stable matrix. The stability of Ac and the boundedness of r imply that the state vector Yc in (6.3.24) is bounded. Since Cc> (sI − Ac )−1 Bc c∗0 = Wm (s), the reference model may be realized by the triple (Ac , Bc c∗0 , Cc ) and described by the nonminimal state space representation Y˙ m = Ac Ym + Bc c∗0 r, ym =

Ym (0) = Ym0

Cc> Ym

(6.3.26)

where Ym ∈ Rnp +2n−2 . Letting e = Yc − Ym to be the state error and e1 = yp − ym the output tracking error, it follows from (6.3.24) and (6.3.26) that e˙ = Ac e,

e1 = Cc> e

i.e., the tracking error e1 satisfies e1 = Cc> eAc t (Yc (0) − Ym (0)) Because Ac is a stable matrix, e1 (t) converges exponentially to zero. The rate of convergence depends on the location of the eigenvalues of Ac , which are equal to the roots of Λ(s)Λ0 (s)Rm (s)Zp (s) = 0. We can affect the rate of convergence by designing Λ(s)Λ0 (s)Rm (s) to have fast zeros, but we are limited by the dependence of Ac on the zeros of Zp (s), which are fixed by the given plant. Example 6.3.1 Let us consider the second order plant yp =

−2 (s + 5) −2 (s + 5) up = 2 up s2 − 2s + 1 (s − 1)

and the reference model ym =

3 r s+3

6.3. MRC FOR SISO PLANTS

341

The order of the plant is np = 2. Its relative degree n∗ = 1 is equal to that of the reference model. We choose the polynomial Λ(s) as Λ(s) = s + 1 = Λ0 (s) and the control input up = θ1∗

1 1 up + θ2∗ yp + θ3∗ yp + c∗0 r s+1 s+1

which gives the closed-loop transfer function yp −2c∗0 (s + 5) (s + 1) = = Gc (s) 2 r (s + 1 − θ1∗ ) (s − 1) + 2 (s + 5) (θ2∗ + θ3∗ (s + 1)) Forcing Gc (s) = 3/(s + 3), we have c∗0 = −3/2 and the matching equation (6.3.14) becomes 2

2

θ1∗ (s − 1) − 2 (θ2∗ + θ3∗ (s + 1)) (s + 5) = (s + 1) (s − 1) − (s + 5) (s + 1) (s + 3) i.e., (θ1∗ − 2θ3∗ ) s2 + (−2θ1∗ − 2θ2∗ − 12θ3∗ ) s + θ1∗ − 10 (θ2∗ + θ3∗ ) = −10s2 − 24s − 14 Equating the powers of s we have θ1∗ − 2θ3∗ + θ2∗ + 6θ3∗ θ1∗ − 10θ2∗ − 10θ3∗ θ1∗

i.e.,



1  1 1

−10 12 −14

 ∗    −2 θ1 −10 6   θ2∗  =  12  −10 θ3∗ −14

0 1 −10

which gives

= = =



   θ1∗ −4  θ2∗  =  −2  θ3∗ 3

The control input is therefore given by up = −4

1 1 up − 2 yp + 3yp − 1.5r s+1 s+1

and is implemented as follows: ω˙ 1 = −ω1 + up ,

ω˙ 2 = −ω2 + yp

up = −4ω1 − 2ω2 + 3yp − 1.5r

5

342

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL r

+ l up +Σ µ ¡ + +£± 6 ¡ ¡ £ ¡ £

- c∗ 0

yp

- Gp (s) ?

?

θ1∗

θ2∗

?

(sI −

-

?

F )−1

(sI − F )−1

g > ¾ ω1 ω2 g> ¾ θ3∗ ¾ Figure 6.6 Implementation of the MRC scheme (6.3.28). Remark 6.3.5 Because θ∗ is a constant vector, the control law (6.3.8) may be also written as up =

α> (s) ∗ α> (s) ∗ (θ1 up ) + (θ yp ) + θ3∗ yp + c∗0 r Λ(s) Λ(s) 2

(6.3.27)

and implemented as ω˙ 1 = F > ω1 + θ1∗ up ω˙ 2 = F > ω2 + θ2∗ yp >

>

up = g ω1 + g ω2 +

(6.3.28) θ3∗ yp

+

c∗0 r

where ω1 , ω2 ∈ Rn−1 ; F, g are as defined in (6.3.23), i.e., g > (sI − > (s) F > )−1 = g > ((sI − F )−1 )> = αΛ(s) . The block diagram for implementing (6.3.28) is shown in Figure 6.6. Remark 6.3.6 The structure of the feedback control law (6.3.8) is not unique. For example, instead of the control law (6.3.8) we can also use α(s) α(s) up = θ1∗> up + θ2∗> yp + c∗0 r (6.3.29) Λ1 (s) Λ1 (s)

6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 343 where Λ1 (s) = Λ0 (s)Zm (s) has degree n, α(s) = αn−1 (s) = [sn−1 , sn−2 , . . ., s, 1]> and θ1∗ , θ2∗ ∈ Rn . The overall desired controller paramh

i>

eter θ∗ = θ1∗> , θ2∗> , c∗0 ∈ R2n+1 has one dimension more than that in (6.3.8). The analysis of (6.3.29) is very similar to that of (6.3.8) and is left as an exercise (see Problem 6.7) for the reader. Remark 6.3.7 We can express the control law (6.3.8) in the general feedback form shown in Figure 6.7 with a feedforward block C(s) =

c∗0 Λ(s) Λ(s) − θ1∗> α(s)

and a feedback block F (s) = −

θ2∗> α(s) + θ3∗ Λ(s) Λ(s)c∗0

where c∗0 , θ1∗ , θ2∗ , θ3∗ are chosen to satisfy the matching equation (6.3.12), (6.3.13). The general structure of Figure 6.7 allows us to analyze and study properties such as robustness, disturbance rejection, etc., of the MRC scheme using well established results from linear system theory [57, 95]. We have shown that the MRC law (6.3.8), whose parameters θi∗ , i = 1, 2, 3 and c∗0 are calculated using the matching equations (6.3.12), (6.3.13) meets the MRC objective. The solution of the matching equations for θi∗ and c∗0 requires the knowledge of the coefficients of the plant polynomials kp Zp (s), Rp (s). In the following sections we combine the MRC law (6.3.8) with an adaptive law that generates estimates for θi∗ , c∗0 on-line to deal with the case of unknown plant parameters.

6.4

Direct MRAC with Unnormalized Adaptive Laws

The main characteristics of the simple MRAC schemes developed in Section 6.2 are (i) The adaptive laws are driven by the estimation error which, due to the special form of the control law, is equal to the regulation or tracking

344

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL r- l C(s) + Σ−

up Gp (s)

yp -

6

F (s) ¾ Figure 6.7 MRC in the general feedback block diagram where C(s) = c∗0 Λ(s) , F (s) Λ(s)−θ1∗> α(s)

=−

θ2∗> α(s)+θ3∗ Λ(s) . c∗0 Λ(s)

error. They are derived using the SPR-Lyapunov design approach without the use of normalization (ii) A simple Lyapunov function is used to design the adaptive law and establish boundedness for all signals in the closed-loop plant. The extension of the results of Sections 6.2, 6.3 to the SISO plant (6.3.1) with unknown parameters became an active area of research in the 70’s. In 1974, Monopoli [150] introduced the concept of the augmented error, a form of an estimation error without normalization, that he used to develop stable MRAC schemes for plants with relative degree 1 and 2, but not for plants with higher relative degree. Following the work of Monopoli, Feuer and Morse [54] designed and analyzed MRAC schemes with unnormalized adaptive laws that are applicable to plants with known relative degree of arbitrary positive value. This generalization came at the expense of additional complexity in the structure of MRAC schemes for plants with relative degree higher than 2. The complexity of the Feuer and Morse MRAC schemes relative to the ones using adaptive laws with normalized estimation errors, introduced during the same period [48, 72, 174], was responsible for their lack of popularity within the adaptive control research community. As a result, it was not until the early 1990s that the MRAC schemes designed for plants with relative degree higher than 2 and employing unnormalized adaptive laws were revived again and shown to offer some advantages over the MRAC with normalization when applied to certain classes of nonlinear plants [98, 116, 117, 118, 162]. In this section, we follow an approach very similar to that of Feuer and Morse and extend the results of Section 6.2 to the general case of higher

6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 345 order SISO plants. The complexity of the schemes increases with the relative degree n∗ of the plant. The simplest cases are the ones where n∗ = 1 and 2. Because of their simplicity, they are still quite popular in the literature of continuous-time MRAC and are presented in separate sections.

6.4.1

Relative Degree n∗ = 1

Let us assume that the relative degree of the plant yp = Gp (s)up = kp

Zp (s) up Rp (s)

(6.4.1)

is n∗ = 1. The reference model ym = Wm (s)r is chosen to have the same relative degree and both Gp (s), Wm (s) satisfy assumptions P1 to P4, and M1 and M2, respectively. In addition Wm (s) is designed to be SPR. The design of the MRAC law to meet the control objective defined in Section 6.3.1 proceeds as follows: We have shown in Section 6.3.2 that the control law ω˙ 1 = F ω1 + gup ,

ω1 (0) = 0

ω˙ 2 = F ω2 + gyp ,

ω2 (0) = 0

up = θ h

∗>

(6.4.2)

ω

i>

h

i>

where ω = ω1> , ω2> , yp , r , and θ∗ = θ1∗> , θ2∗> , θ3∗ , c∗0 calculated from the matching equation (6.3.12) and (6.3.13) meets the MRC objective defined in Section 6.3.1. Because the parameters of the plant are unknown, the desired controller parameter vector θ∗ cannot be calculated from the matching equation and therefore (6.4.2) cannot be implemented. A reasonable approach to follow in the unknown plant parameter case is to replace (6.4.2) with the control law ω˙ 1 = F ω1 + gup ,

ω1 (0) = 0

ω˙ 2 = F ω2 + gyp ,

ω2 (0) = 0

>

up = θ ω

(6.4.3)

346

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

where θ(t) is the estimate of θ∗ at time t to be generated by an appropriate adaptive law. We derive such an adaptive law by following a similar procedure as in the case of the examples of Section 6.2. We first obtain a composite state space representation of the plant and controller, i.e., Y˙ c = A0 Yc + Bc up , yp =

Yc (0) = Y0

Cc> Yc >

up = θ ω h

> > where Yc = x> p , ω1 , ω2

i>

.



A0

Ap 0  F =  0 > gCp 0

Cc> =

h

Cp> , 0, 0

i







0 Bp    0  , Bc =  g  F 0

and then add and subtract the desired input Bc θ∗> ω to obtain ³

Y˙ c = A0 Yc + Bc θ∗> ω + Bc up − θ∗> ω

´

If we now absorb the term Bc θ∗> ω into the homogeneous part of the above equation, we end up with the representation ³

´

Y˙ c = Ac Yc + Bc c∗0 r + Bc up − θ∗> ω , Yc (0) = Y0 yp = Cc> Yc

(6.4.4)

where Ac is as defined in (6.3.25). Equation (6.4.4) is the same as the closedloop equation (6.3.24) in the known parameter case except for the additional input term Bc (up − θ∗> ω) that depends on the choice of the input up . It serves as the parameterization of the plant equation in terms of the desired controller parameter vector θ∗ . Let e = Yc − Ym and e1 = yp − ym where Ym is the state of the nonminimal representation of the reference model given by (6.3.26), we obtain the error equation e˙ = Ac e + Bc (up − θ∗> ω), e1 =

Cc> e

e(0) = e0 (6.4.5)

6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 347 Because Cc> (sI − Ac )−1 Bc c∗0 = Wm (s) we have

³

e1 = Wm (s)ρ∗ up − θ∗> ω

´

(6.4.6)

where ρ∗ = c1∗ , which is in the form of the bilinear parametric model analyzed 0 in Chapter 4. We can now use (6.4.6) to generate a wide class of adaptive laws for estimating θ∗ by using the results of Chapter 4. We should note that (6.4.5) and (6.4.6) hold for any relative degree and will also be used in later sections. The estimate eˆ1 (t) of e1 (t) based on θ(t), the estimate of θ∗ at time t, is given by ³ ´ eˆ1 = Wm (s)ρ up − θ> ω (6.4.7) where ρ is the estimate of ρ∗ . Because the control input is given by up = θ> (t)ω it follows that eˆ1 = Wm (s)[0]; therefore, the estimation error ²1 defined in Chapter 4 as ²1 = e1 − eˆ1 may be taken to be equal to e1 , i.e., ²1 = e1 . Consequently, (6.4.7) is not needed and the estimate ρ of ρ∗ does not have to be generated. Substituting for the control law in (6.4.5), we obtain the error equation ¯c ρ∗ θ˜> ω, e(0) = e0 e˙ = Ac e + B e1 = Cc> e

(6.4.8)

where ¯c = Bc c∗ B 0 or e1 = Wm (s)ρ∗ θ˜> ω 4 which relates the parameter error θ˜ = θ(t) − θ∗ with the tracking error e1 . Because Wm (s) = Cc> (sI − Ac )−1 Bc c∗0 is SPR and Ac is stable, equation (6.4.8) is in the appropriate form for applying the SPR-Lyapunov design approach.

348

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL We therefore proceed by proposing the Lyapunov-like function > ³ ´ ˜> −1 ˜ ˜ e = e Pc e + θ Γ θ |ρ∗ | V θ, 2 2

(6.4.9)

where Γ = Γ> > 0 and Pc = Pc> > 0 satisfies the algebraic equations > Pc Ac + A> c Pc = −qq − νc Lc ¯c = Cc Pc B

where q is a vector, Lc = L> c > 0 and νc > 0 is a small constant, that are implied by the MKY lemma. The time derivative V˙ of V along the solution of (6.4.8) is given by e> qq > e νc > ˜˙ ∗ | ¯c ρ∗ θ˜> ω + θ˜> Γ−1 θ|ρ V˙ = − − e Lc e + e> Pc B 2 2 ¯c = e1 and ρ∗ = |ρ∗ |sgn(ρ∗ ), we can make V˙ ≤ 0 by choosing Because e> Pc B

which leads to

˙ θ˜ = θ˙ = −Γe1 ω sgn(ρ∗ )

(6.4.10)

e> qq > e νc > V˙ = − − e Lc e 2 2

(6.4.11)

Equations (6.4.9) and (6.4.11) imply that V and, therefore, e, θ˜ ∈ L∞ . Because e = Yc −Ym and Ym ∈ L∞ , we have Yc ∈ L∞ , which implies that yp , ω1 , ω2 ∈ L∞ . Because up = θ> ω and θ, ω ∈ L∞ we also have up ∈ L∞ . Therefore all the signals in the closed-loop plant are bounded. It remains to show that the tracking error e1 = yp − ym goes to zero as t → ∞. From (6.4.9) and (6.4.11) we establish that e and therefore e1 ∈ L2 . Furthermore, using θ, ω, e ∈ L∞ in (6.4.8) we have that e, ˙ e˙ 1 ∈ L∞ . Hence, e1 , e˙ 1 ∈ L∞ and e1 ∈ L2 , which, by Lemma 3.2.5, imply that e1 (t) → 0 as t → ∞. We summarize the main equations of the MRAC scheme in Table 6.1. The stability properties of the MRAC scheme of Table 6.1 are given by the following theorem. Theorem 6.4.1 The MRAC scheme summarized in Table 6.1 guarantees that:

6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 349 Table 6.1 MRAC scheme: n∗ = 1 Z (s)

Plant

yp = kp Rpp (s) up , n∗ = 1

Reference model

Zm (s) ym = Wm (s)r, Wm (s) = km R m (s)

Control law

ω˙ 1 = F ω1 + gup , ω1 (0) = 0 ω˙ 2 = F ω2 + gyp , ω2 (0) = 0 up = θ> ω ω = [ω1> , ω2> , yp , r]> , ω1 ∈ Rn−1 , ω2 ∈ Rn−1

Adaptive law

θ˙ = −Γe1 ω sgn(ρ∗ ) e1 = yp − ym , sgn(ρ∗ ) = sgn(kp /km )

Assumptions

Zp , Rp and Wm (s) satisfy assumptions P1 to P4, and M1 and M2, respectively; Wm (s) is SPR; (sI − α(s) , α(s) = [sn−2 , sn−3 , . . . s, 1]> , where F )−1 g = Λ(s) Λ = Λ0 Zm is Hurwitz, and Λ0 (s) is of degree n−1−qm, qm is the degree of Zm(s); Γ = Γ> > 0 is arbitrary

(i) All signals in the closed-loop plant are bounded and the tracking error e1 converges to zero asymptotically with time for any reference input r ∈ L∞ . (ii) If r is sufficiently rich of order 2n, r˙ ∈ L∞ and Zp (s), Rp (s) are rela˜ = |θ − θ∗ | and the tracking tively coprime, then the parameter error |θ| error e1 converge to zero exponentially fast. Proof (i) This part has already been completed above. (ii) Equations (6.4.8) and (6.4.10) have the same form as (4.3.30) and (4.3.35) with n2s = 0 in Chapter 4 whose convergence properties are established by Corollary

350

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

4.3.1. Therefore, by using the same steps as in the proof of Corollary 4.3.1 we can ˜ → 0 exponentially fast. If r˙ ∈ L∞ establish that if ω, ω˙ ∈ L∞ and ω is PE then θ(t) then it follows from the results of part (i) that ω˙ ∈ L∞ . For the proof to be complete, it remains to show that ω is PE. We express ω as   (sI − F )−1 gG−1 p (s)yp   (sI − F )−1 gyp  ω= (6.4.12)   yp r Because yp = ym + e1 = Wm (s)r + e1 we have ω = ωm + ω ¯

(6.4.13)

where ωm = H(s)r, and

ω ¯ = H0 (s)e1

  (sI − F )−1 gG−1 (sI − F )−1 gG−1 p (s) p (s)Wm (s) −1 −1    (sI − F ) g (sI − F ) gW (s) m  , H0 (s) =  H(s) =     Wm (s) 1 1 0 

   

The vector ω ¯ is the output of a proper transfer matrix whose poles are stable and whose input e1 ∈ L2 ∩ L∞ and goes to zero as t → ∞. Hence, from Corollary 3.3.1 we have ω ¯ ∈ L2 ∩ L∞ and |¯ ω (t)| → 0 as t → ∞. It then follows from Lemma 4.8.3 that ω is PE if ωm is PE. It remains to show that ωm is PE when r is sufficiently rich of order 2n. Because r is sufficiently rich of order 2n, according to Theorem 5.2.1, we can show that ωm is PE by proving that H(jω1 ), H(jω2 ), . . . , H(jω2n ) are linearly independent on C 2n for any ω1 , ω2 , . . . , ω2n ∈ R with ωi 6= ωj for i 6= j. From the definition of H(s), we can write   α(s)Rp (s)km Zm (s)  α(s)kp Zp (s)km Zm (s)  4 1 1  = H1 (s) H(s) = kp Zp (s)Λ(s)Rm (s)  Λ(s)kp Zp (s)km Zm (s)  kp Zp (s)Λ(s)Rm (s) Λ(s)kp Zp (s)Rm (s) (6.4.14) Because all the elements of H1 (s) are polynomials of s with order less than or equal to that of Λ(s)Zp (s)Rm (s), we can write  l  s  sl−1   ¯ H1 (s) = H (6.4.15)  ..   .  1

6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 351 4

where l = 2n − 1 + qm is the order of the polynomial Λ(s)Zp (s)Rm (s), qm is the ¯ ∈ R2n×(l+1) is a constant matrix. degree of Zm (s) and H ¯ in (6.4.15) is of full rank, i.e., rank(H) ¯ = We now prove by contradiction that H 2n ¯ 2n. Suppose rank(H) < 2n, i.e., there exists a constant vector C ∈ R with C 6= 0 such that ¯ =0 C >H or equivalently C > H1 (s) = 0

(6.4.16)

for all s ∈ C. Let C = [C1> , C2> , c3 , c4 ]> , where C1 , C2 ∈ Rn−1 , c3 , c4 ∈ R1 , then (6.4.16) can be written as C1> α(s)Rp (s)km Zm (s) +c3 Λ(s)kp Zp (s)km Zm (s)

+ C2> α(s)kp Zp (s)km Zm (s) + c4 Λ(s)Rm (s)kp Zp (s) = 0

(6.4.17)

Because the leading coefficient of the polynomial on the left hand side is c4 , for (6.4.17) to hold, it is necessary that c4 = 0. Therefore, [C1> α(s)Rp (s) + C2> α(s)kp Zp (s) + c3 Λ(s)kp Zp (s)]km Zm (s) = 0 or equivalently C1> α(s)Rp (s) + C2> α(s)kp Zp (s) + c3 Λ(s)kp Zp (s) = 0

(6.4.18)

Equation (6.4.18) implies that kp

Zp (s) C1> α(s) =− Rp (s) c3 Λ(s) + C2> α(s)

(6.4.19)

Noting that c3 Λ(s) + C2> α(s) is of order at most equal to n − 1, (6.4.19) contradicts ¯ must be of full rank. our assumption that Zp (s), Rp (s) are coprime. Therefore H 4

Now consider the 2n × 2n matrix L(ω1 , . . . , ω2n ) = [H(jω1 ), , . . . , H(jω2n )]. Using (6.4.14) and (6.4.15), we can express L(ω1 , ω2 , . . . , ω2n ) as   (jω1 )l (jω2 )l ... (jω2n )l  (jω1 )l−1 (jω2 )l−1 . . . (jω2n )l−1   ¯ L(ω1 , . . . , ω2n ) = H   .. .. ..   . . .    ×  

1 1 D(jω1 )

0 0 0

1 0 1 D(jω2 )

0

... ... ... .. . ···

1 0 0 0

1 D(jω2n )

     

(6.4.20)

352

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

where D(s) = kp Zp (s)Λ(s)Rm (s). Note that the matrix in the middle of the righthand side of (6.4.20) is a submatrix of the Vandermonte matrix, which is always nonsingular for ωi 6= ωk , i 6= k; i, k = 1, . . . , 2n. We, therefore, conclude from (6.4.20) that L(ω1 , . . . , ω2n ) is of full rank which implies that H(jω1 ), . . . , H(jω2n ) are linearly independent on C 2n and the proof is complete. 2

Example 6.4.1 Let us consider the second order plant yp =

kp (s + b0 ) up (s2 + a1 s + a0 )

where kp > 0, b0 > 0 and kp , b0 , a1 , a0 are unknown constants. The desired performance of the plant is specified by the reference model ym =

1 r s+1

Using Table 6.1, the control law is designed as ω˙ 1 ω˙ 2 up

= = =

−2ω1 + up , ω1 (0) = 0 −2ω2 + yp , ω2 (0) = 0 θ1 ω1 + θ2 ω2 + θ3 yp + c0 r

by choosing F = −2, g = 1 and Λ(s) = s + 2. The adaptive law is given by θ˙ = −Γe1 ω,

θ(0) = θ0

>

>

where e1 = yp − ym , θ = [θ1 , θ2 , θ3 , c0 ] and ω = [ω1 , ω2 , yp , r] . We can choose Γ = diag{γi } for some γi > 0 and obtain the decoupled adaptive law θ˙i = −γi e1 ωi , i = 1, . . . , 4 where θ4 = c0 , ω3 = yp , ω4 = r; or we can choose Γ to be any positive definite matrix. For parameter convergence, we choose r to be sufficiently rich of order 4. As an example, we select r = A1 sin ω1 t + A2 sin ω2 t for some nonzero constants A1 , A2 , ω1 , ω2 with ω1 6= ω2 . We should emphasize that we may not always have the luxury to choose r to be sufficiently rich. For example, if the control objective requires r =constant in order for yp to follow a constant set point at steady state, then the use of a sufficiently rich input r of order 4 will destroy the desired tracking properties of the closed-loop plant. The simulation results for the MRAC scheme for the plant with b0 = 3, a1 = 3, a0 = −10, kp = 1 are shown in Figures 6.8 and 6.9. The initial value of the

6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 353

tracking error

3 2 1 0 -1 0

2

4

6

8

10 (a)

12

14

16

18

20 sec

2

4

6

8

10 (b)

12

14

16

18

20 sec

parameter error

2 1 0 -1 -2 -3 0

Figure 6.8 Response of the MRAC scheme for Example 6.4.1 with r(t) = unit step function.

parameters are chosen as θ(0) = [3, −10, 2, 3]> . Figure 6.8(a, b) shows the response of the tracking error e1 and estimated parameter error θ˜i for γi = 1 and r = unit step . Figure 6.9 shows the simulation results for Γ = diag{2, 6, 6, 2} and r = 0.5 sin 0.7t + 2 cos 5.9t. From Figure 6.9 (b), we note that the estimated parameters converge to θ∗ = [1, −12, 0, 1]> due to the use of a sufficiently rich input. 5

Remark 6.4.1 The error equation (6.4.8) takes into account the initial conditions of the plant states. Therefore the results of Theorem 6.4.1 hold

354

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

tracking error

2

1

0

-1 0

20

40

60

80

100 (a)

120

140

160

180

200 sec

20

40

60

80

100 (b)

120

140

160

180

200 sec

parameter error

2 0 -2 -4 0

Figure 6.9 Response of the MRAC scheme for Example 6.4.1 with r(t) = 0.5 sin 0.7t + 2 cos 5.9t. for any finite initial condition for the states of the plant and filters. In the analysis, we implicitly assumed that the nonlinear differential ˜ = equations (6.4.8) and (6.4.10) with initial conditions e(0) = e0 , θ(0) ˜ θ0 possess a unique solution. For our analysis to be valid, the solution ˜ e(t) has to exist for all t ∈ [0, ∞). The existence and uniqueness θ(t), of solutions of adaptive control systems is addressed in [191]. Remark 6.4.2 The proof of Theorem 6.4.1 part (i) may be performed by using a minimal state-space representation for the equation e1 = Wm (s)ρ∗ θ˜> ω

(6.4.21)

6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 355 rather than the nonminimal state representation (6.4.8) to develop the adaptive law θ˙ = −Γe1 ω. In this case we establish that e1 , θ ∈ L∞ and e1 ∈ L2 by using the LKY ( instead of the MKY Lemma ) and the properties of a Lyapunov-like function. The boundedness of e1 implies that yp ∈ L∞ . The boundedness of ω and the rest of the signals requires the following additional arguments: We write ω as    

ω=

(sI − F )−1 gG−1 p (s)yp −1 (sI − F ) gyp yp r

    

−1 Because yp ∈ L∞ and (sI − F )−1 gG−1 p (s), (sI − F ) g are proper (note that the relative degree of Gp (s) is 1) with stable poles, we have ω ∈ L∞ . From up = θ> ω and θ, ω ∈ L∞ , it follows that up ∈ L∞ . The proof of e1 (t) → 0 as t → ∞ follows by applying Lemma 3.2.5 and using the properties e1 ∈ L2 , e˙ 1 = sWm (s)ρ∗ θ˜> ω ∈ L∞ .

Remark 6.4.3 The effect of initial conditions may be accounted for by considering e1 = Wm (s)ρ∗ θ˜> ω + Cc> (sI − Ac )−1 e(0) instead of (6.4.21). Because the term that depends on e(0) is exponentially decaying to zero, it does not affect the stability results. This can be shown by modifying the Lyapunov-like function to accommodate the exponentially decaying to zero term (see Problem 6.19).

6.4.2

Relative Degree n∗ = 2

Let us consider again the parameterization of the plant in terms of θ∗ , developed in the previous section, i.e., ³

e˙ = Ac e + Bc up − θ∗> ω e1 = Cc> e

´

(6.4.22)

356

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

or

³

e1 = Wm (s)ρ∗ up − θ∗> ω

´

(6.4.23)

In the relative degree n∗ = 1 case, we are able to design Wm (s) to be SPR which together with the control law up = θ> ω enables us to obtain an error equation that is suitable for applying the SPR-Lyapunov design method. With n∗ = 2, Wm (s) can no longer be designed to be SPR and therefore the procedure of Section 6.4.1 fails to apply here. Instead, let us follow the techniques of Chapter 4 and use the identity (s + p0 )(s + p0 )−1 = 1 for some p0 > 0 to rewrite (6.4.22), (6.4.23) as ³

´

¯c (s + p0 ) ρ∗ uf − θ∗> φ , e˙ = Ac e + B

e(0) = e0

e1 = Cc> e

(6.4.24)

i.e.,

³

´

e1 = Wm (s) (s + p0 ) ρ∗ uf − θ∗> φ

(6.4.25)

¯c = Bc c∗ , where B 0

1 1 up , φ = ω s + p0 s + p0 and Wm (s), p0 > 0 are chosen so that Wm (s) (s + p0 ) is SPR. We use ρ, θ, the estimate of ρ∗ , θ∗ , respectively, to generate the estimate eˆ1 of e1 as eˆ1 = Wm (s)(s + p0 )ρ(uf − θ> φ) uf =

If we follow the same procedure as in Section 6.4.2, then the next step is to choose up so that eˆ1 = Wm (s)(s + p0 )[0], ²1 = e1 and (6.4.24) is in the form of the error equation (6.4.8) where the tracking error e1 is related to the parameter error θ˜ through an SPR transfer function. The control law up = θ> ω (used in the case of n∗ = 1 ) motivated from the known parameter case cannot transform (6.4.23) into the error equation we are looking for. Instead if we choose up so that uf = θ > φ

(6.4.26)

we have eˆ1 = Wm (s)(s + p0 )[0] and by substituting (6.4.26) in (6.4.24), we obtain the error equation ¯c (s + p0 ) ρ∗ θ˜> φ, e˙ = Ac e + B e1 =

Cc> e

e(0) = e0 (6.4.27)

6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 357 or in the transfer function form e1 = Wm (s) (s + p0 ) ρ∗ θ˜> φ which can be transformed into the desired form by using the transformation ¯c ρ∗ θ˜> φ e¯ = e − B

(6.4.28)

i.e., e¯˙ = Ac e¯ + B1 ρ∗ θ˜> φ, e¯(0) = e¯0 e1 = Cc> e¯

(6.4.29)

¯c + B ¯c p0 and Cc> B ¯c = Cp> Bp c∗ = 0 due to n∗ = 2. With where B1 = Ac B 0 (6.4.29), we can proceed as in the case of n∗ = 1 and develop an adaptive law for θ. Let us first examine whether we can choose up to satisfy equation (6.4.26). We have up = (s + p0 ) uf = (s + p0 ) θ> φ which implies that up = θ> ω + θ˙> φ

(6.4.30)

Because θ˙ is made available by the adaptive law, the control law (6.4.30) can be implemented without the use of differentiators. Let us now go back to the error equation (6.4.29). Because ¯c (s + p0 ) = Wm (s) (s + p0 ) Cc> (sI − Ac )−1 B1 = Cc> (sI − Ac )−1 B is SPR, (6.4.29) is of the same form as (6.4.8) and the adaptive law for generating θ is designed by considering > ³ ´ ˜> −1 ˜ ˜ e¯ = e¯ Pc e¯ + θ Γ θ |ρ∗ | V θ, 2 2

where Pc = Pc> > 0 satisfies the MKY Lemma. As in the case of n∗ = 1, for ˙ θ˜ = θ˙ = −Γe1 φ sgn(ρ∗ )

(6.4.31)

the time derivative V˙ of V along the solution of (6.4.29), (6.4.31) is given by e¯> qq > e¯ νc > V˙ = − − e¯ Lc e¯ ≤ 0 2 2

358

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

˜ e1 ∈ L∞ and e¯, e1 ∈ L2 . Because e1 = yp − ym , we which implies that e¯, θ, also have yp ∈ L∞ . The signal vector φ is expressed as 

φ=

1 s + p0

   

(sI − F )−1 gG−1 p (s)yp (sI − F )−1 gyp yp r

    

(6.4.32)

by using up = G−1 p (s)yp . We can observe that each element of φ is the output of a proper stable transfer function whose input is yp or r. Because yp , r ∈ L∞ we have φ ∈ L∞ . Now e¯, θ, φ ∈ L∞ imply (from (6.4.28) ) that e and, therefore, Yc ∈ L∞ . Because ω, φ, e1 ∈ L∞ we have θ˙ ∈ L∞ and up ∈ L∞ and therefore all signals in the closed -loop plant are bounded. From (6.4.29) we also have that e¯˙ ∈ L∞ , i.e., e˙ 1 ∈ L∞ , which, together with e1 ∈ L∞ ∩ L2 , implies that e1 (t) → 0 as t → ∞. We present the main equations of the overall MRAC scheme in Table 6.2 and summarize its stability properties by the following theorem. Theorem 6.4.2 The MRAC scheme of Table 6.2 guarantees that (i) All signals in the closed-loop plant are bounded and the tracking error e1 converges to zero asymptotically. (ii) If Rp , Zp are coprime and r is sufficiently rich of order 2n, then the ˜ = |θ − θ∗ | and the tracking error e1 converge to parameter error |θ| zero exponentially fast. Proof (i) This part has been completed above. (ii) Consider the error equations (6.4.29), (6.4.31) which have the same form as equations (4.3.10), (4.3.35) with n2s = 0 in Chapter 4. Using Corollary 4.3.1 we have ˜ that if φ, φ˙ ∈ L∞ and φ is PE, then the adaptive law (6.4.31) guarantees that |θ| converges to zero exponentially fast. We have already established that yp , φ ∈ L∞ . It follows from (6.4.32) and the fact that e˙ 1 and therefore y˙ p ∈ L∞ that φ˙ ∈ L∞ . Hence it remains to show that φ is PE. As in the case of n∗ = 1 we write φ as φ = φm + φ¯

6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 359 Table 6.2 MRAC scheme: n∗ = 2 Z (s)

Plant

yp = kp Rpp (s) up , n∗ = 2

Reference model

Zm (s) ym = Wm (s)r, Wm (s) = km R m (s)

ω˙ 1 = F ω1 + gup , ω1 (0) = 0 ω˙ 2 = F ω2 + gyp , ω2 (0) = 0 φ˙ = −p0 φ + ω, φ(0) = 0 up = θ> ω + θ˙> φ = θ> ω − φ> Γφe1 sgn(kp /km ) ω = [ω1> , ω2> , yp , r]> , ω1 ∈ Rn−1 , ω2 ∈ Rn−1

Control law

Adaptive law

θ˙ = −Γe1 φsgn(kp /km ),

Assumptions

Zp (s) is Hurwitz; Wm (s)(s +p0 ) is strictly proper and SPR; F, g, Γ are as defined in Table 6.1; plant and reference model satisfy assumptions P1 to P4, and M1 and M2, respectively

where



φm

and

e1 = yp − ym

 (sI − F )−1 gG−1 p (s)Wm (s)  1  (sI − F )−1 gWm (s)  r =   Wm (s) s + p0 1 

 (sI − F )−1 gG−1 p (s)  1  (sI − F )−1 g   e1 φ¯ =   1 s + p0 0

Because e1 ∈ L2 ∩ L∞ and e1 → 0 as t → ∞ it follows (see Corollary 3.3.1) that ¯ → 0 as t → ∞. φ¯ ∈ L2 ∩ L∞ and |φ|

360

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

Proceeding as in the proof of Theorem 6.4.1, we establish that φm is PE and use Lemma 4.8.3 to show that φ is also PE which implies, using the results of Chapter 4, ˜ → 0 exponentially fast. Using (6.4.29) and the exponential convergence of that |θ| ˜ |θ| to zero we obtain that e1 converges to zero exponentially fast. 2 Example 6.4.2 Let us consider the second order plant yp =

kp up (s2 + a1 s + a0 )

where kp > 0, and a1 , a0 are constants. The reference model is chosen as ym =

5 (s + 5)

2r

Using Table 6.2 the control law is designed as ω˙ 1 ω˙ 2 φ˙ up

= −2ω1 + up = −2ω2 + yp = −φ + ω = θ> ω − φ> Γφe1

>

where ω = [ω1 , ω2 , yp , r] , e1 = yp − ym , p0 = 1, Λ(s) = s + 2 and The adaptive law is given by θ˙ = −Γe1 φ

5(s+1) (s+5)2

is SPR.

where Γ = Γ> > 0 is any positive definite matrix and θ = [θ1 , θ2 , θ3 , c0 ]> . For parameter convergence, the input up is chosen as up = A1 sin ω1 t + A2 sin ω2 t for some A1 , A2 6= 0 and ω1 6= ω2 . Figures 6.10 and 6.11 show some simulation results of the MRAC scheme for the plant with a1 = 3, a0 = −10, kp = 1. We start with an initial parameter vector θ(0) = [3, 18, −8, 3]> that leads to an initially destabilizing controller. The tracking error and estimated parameter error response is shown in Figure 6.10 for Γ = diag{2, 4, 0.8, 1}, and r = unit step. Due to the initial destabilizing controller, the transient response is poor. The adaptive mechanism alters the unstable behavior of the initial controller and eventually drives the tracking error to zero. In Figure 6.11 we show the response of the same system when r = 3 sin 4.9t + 0.5 cos 0.7t is a sufficiently rich input of order 4. Because of the use of a sufficiently rich signal, θ(t) converges to θ∗ = [1, 12, −10, 1]> . 5

6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 361

tracking error

5

0

-5 0

5

10

15

20

25 (a)

30

35

40

45

50 sec

5

10

15

20

25 (b)

30

35

40

45

50 sec

parameter error

6 4 2 0 -2 -4 0

Figure 6.10 Response of the MRAC scheme for Example 6.4.2 with r(t) = unit step function.

Remark 6.4.4 The control law (6.4.30) is a modification of the certainty equivalence control law up = θ> ω and is motivated from stability considerations. The additional term θ˙> φ = −φ> Γφe1 sgn(ρ∗ ) is a nonlinear one that disappears asymptotically with time, i.e., up = θ> ω + θ˙> φ converges to the certainty equivalence control law as t → ∞. The number and complexity of the additional terms in the certainty equivalence control law increase with the relative degree n∗ as we demonstrate in the next section.

362

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

tracking error

5

0

-5 0

5

10

15

20

25 (a)

30

35

40

45

50 sec

5

10

15

20

25 (b)

30

35

40

45

50 sec

parameter error

6 4 2 0 -2 -4 0

Figure 6.11 Response of the MRAC scheme for Example 6.4.2 with r(t) = 3 sin 4.9t + 0.5 sin 0.7t.

Remark 6.4.5 The proof of Theorem 6.4.2 may be accomplished by using a minimal state space realization for the error equation e1 = Wm (s)(s + p0 )ρ∗ θ˜> φ The details of such an approach are left as an exercise for the reader.

6.4.3

Relative Degree n∗ = 3

As in the case of n∗ = 2, the transfer function Wm (s) of the reference model cannot be chosen to be SPR because according to assumption (M2),

6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 363 Wm (s) should have the same relative degree as the plant transfer function. Therefore, the choice of up = θ> ω in the error equation e1 = Wm (s)ρ∗ (up − θ∗> ω)

(6.4.33)

will not lead to the desired error equation where the tracking error is related to the parameter error through an SPR transfer function. As in the case of n∗ = 2, let us rewrite (6.4.33) in a form that involves an SPR transfer function by using the techniques of Chapter 4, i.e., we express (6.4.33) as ³

´

e1 = Wm (s)(s + p0 )(s + p1 )ρ∗ uf − θ∗> φ where uf =

(6.4.34)

1 1 up , φ = ω (s + p0 )(s + p1 ) (s + p0 )(s + p1 )

4 ¯ m (s) = and Wm (s), p0 , p1 are chosen so that W Wm (s)(s + p0 )(s + p1 ) is ¯ m (s) is 1. For SPR, which is now possible because the relative degree of W simplicity and without loss of generality let us choose

1 (s + p0 )(s + p1 )(s + q0 )

Wm (s) = for some q0 > 0 so that e1 =

1 ρ∗ (uf − θ∗> φ) s + q0

(6.4.35)

The estimate of eˆ1 of e1 based on the estimates ρ, θ is given by eˆ1 =

1 ρ(uf − θ> φ) s + q0

(6.4.36)

If we proceed as in the case of n∗ = 2 we would attempt to choose uf = θ > φ to make eˆ1 =

1 s+q0 [0]

(6.4.37)

and obtain the error equation e1 =

1 ρ∗ θ˜> φ s + q0

(6.4.38)

364

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

The adaptive law θ˙ = −Γe1 φsgn(ρ∗ )

(6.4.39)

will then follow by using the standard procedure. Equation (6.4.37), however, implies the use of the control input up = (s + p0 )(s + p1 )uf = (s + p0 )(s + p1 )θ> φ

(6.4.40)

¨ that is not available for measurement. Consequently the which involves θ, control law (6.4.40) cannot be implemented and the choice of uf = θ> φ is not feasible. The difficulty of not being able to extend the results for n∗ = 1, 2 to n∗ ≥ 3 became the major obstacle in advancing research in adaptive control during the 1970s. By the end of the 1970s and early 1980s, however, this difficulty was circumvented and several successful MRAC schemes were proposed using different approaches. Efforts to extend the procedure of n∗ = 1, 2 to n∗ ≥ 3 continued during the early 1990s and led to new designs for MRAC . One such design proposed by Morse [164] employs the same control law as in (6.4.40) but the adaptive law for θ is modified in such a way that θ¨ becomes an available signal. This modification, achieved at the expense of a higher-order adaptive law, led to a MRAC scheme that guarantees signal boundedness and convergence of the tracking error to zero. Another successful MRAC design that has it roots in the paper of Feuer and Morse [54] is proposed in [162] for a third order plant with known high frequency gain. In this design, the adaptive law is kept unchanged but the control law is chosen as up = θ> ω + ua where ua is designed based on stability considerations. Below we present and analyze a very similar design as in [162]. We start by rewriting (6.4.35), (6.4.36) as e1 =

1 ρ∗ (θ˜> φ + r0 ), s + q0

eˆ1 =

1 ρr0 s + q0

(6.4.41)

where r0 = uf − θ> φ and θ˜ = θ − θ∗ . Because r0 cannot be forced to be equal to zero by setting uf = θ> φ, we will focus on choosing up so that r0 goes to zero as t → ∞. In this case,

6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 365 the estimation error ²1 = e1 − eˆ1 is not equal to e1 because eˆ1 6= 0 owing to ρr0 6= 0. However, it satisfies the error equation ²1 = e1 − eˆ1 =

1 (ρ∗ θ˜> φ − ρ˜r0 ) s + q0

(6.4.42)

that leads to the adaptive law θ˙ = −Γ²1 φsgn(ρ∗ ), ρ˙ = γ²1 r0

(6.4.43)

where Γ = Γ> and γ > 0 by considering the Lyapunov-like function V =

²21 θ˜> Γ−1 θ˜ ∗ ρ˜2 + |ρ | + 2 2 2γ

We now need to choose ua in up = θ> ω + ua to establish stability for the system (6.4.41) to (6.4.43). Let us now express r0 as r0 = uf − θ> φ = where u1 =

i 1 h u1 − θ˙> φ − θ> φ1 s + p0

1 1 up , φ1 = (s + p0 )φ = ω s + p1 s + p1

i.e., r˙0 = −p0 r0 + u1 − θ˙> φ − θ> φ1

(6.4.44)

˙ we obtain Substituting for θ, r˙0 = −p0 r0 + u1 + φ> Γφ²1 sgn(ρ∗ ) − θ> φ1

(6.4.45)

If we now choose u1 = −φ> Γφ²1 sgn(ρ∗ ) + θ> φ1 then r˙0 = −p0 r0 and r0 converges to zero exponentially fast. This choice of u1 , however, leads to a control input up that is not implementable since up = (s + p1 )u1 will involve the first derivative of u1 and, therefore, the derivative of e1 that is not available for measurement. Therefore, the term φ> Γφ²1 sgn(ρ∗ ) in (6.4.45) cannot be eliminated by u1 . Its effect, however, may be counteracted by introducing what is called a “nonlinear damping” term in u1 [99]. That is, we choose ³ ´2 u1 = θ> φ1 − α0 φ> Γφ r0 (6.4.46)

366

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

where α0 > 0 is a design constant, and obtain ·

´2 ¸

³

r˙0 = − p0 + α0 φ> Γφ

r0 + φ> Γφ²1 sgn(ρ∗ )

The purpose of the nonlinear term (φ> Γφ)2 is to “damp out” the possible destabilizing effect of the nonlinear term φ> Γφ²1 as we show in the analysis to follow. Using (6.4.46), the control input up = (s + p1 )u1 is given by up = θ> ω + θ˙> φ1 − (s + p1 )α0 (φ> Γφ)2 r0

(6.4.47)

If we now perform the differentiation in (6.4.47) and substitute for the derivative of r0 we obtain ³

´

³

´2

up = θ> ω + θ˙> φ1 − 4α0 φ> Γφ φ> Γφ˙ r0 − α0 (p1 − p0 ) φ> Γφ ³

´4

+α02 φ> Γφ

³

´3

r0 − α0 φ> Γφ

²1 sgn(ρ∗ )

r0

(6.4.48)

where φ˙ is generated from φ˙ =

s ω (s + p0 )(s + p1 )

which demonstrates that up can be implemented without the use of differentiators. We summarize the main equations of the MRAC scheme in Table 6.3. The stability properties of the proposed MRAC scheme listed in Table 6.3 are summarized as follows. Theorem 6.4.3 The MRAC scheme of Table 6.3 guarantees that (i) All signals in the closed-loop plant are bounded and r0 (t), e1 (t) → 0 as t → ∞. (ii) If kp is known, r is sufficiently rich of order 2n and Zp , Rp are coprime, ˜ = |θ − θ∗ | and tracking error e1 converge then the parameter error |θ| to zero exponentially fast. ˜ and (iii) If r is sufficiently rich of order 2n and Zp , Rp are coprime, then |θ| e1 converge to zero asymptotically (not necessarily exponentially fast). (iv) The estimate ρ converges to a constant ρ¯ asymptotically independent of the richness of r.

6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 367 Table 6.3 MRAC scheme: n∗ = 3

Z (s)

n∗ = 3

Plant

yp = kp Rpp (s) up ,

Reference model

ym = Wm (s)r

Control law

ω˙ 1 = F ω1 + gup , ω1 (0) = 0 ω˙ 2 = F ω2 + gyp , ω2 (0) = 0 r˙0 = −(p0 + α0 (φ> Γφ)2 )r0 + φ> Γφ²1 sgn(ρ∗ ) up = θ> ω + ua ˙ 0 ua = θ˙>φ1−α0 (p1−p0 )(φ>Γφ)2 r0−4α0 φ> Γφ(φ> Γφ)r 2 > 4 > 3 ∗ +α0 (φ Γφ) r0 − α0 (φ Γφ) ²1 sgn(ρ )

Adaptive law

Design variables

θ˙ = −Γ²1 φsgn(ρ∗ ), ρ˙ = γ²1 r0 1 ²1 = e1 − eˆ1 , eˆ1 = s+q ρr0 0 h

1 φ = (s+p0 )(s+p ω, ω = ω1> , ω2> , yp , r 1) 1 ω, e1 = yp − ym φ1 = s+p 1

i>

Γ = Γ> > 0, γ > 0, α0 > 0 are arbitrary design constants; Wm (s)(s + p0 )(s + p1 ) is strictly proper and SPR; F, g are as in the case of n∗ = 1; Zp (s), Rp (s) and Wm (s) satisfy assumptions P1 to P4, M1 and M2, respectively; sgn(ρ∗ ) = sgn(kp /km )

Proof (i) The equations that describe the stability properties of the closed-loop plant are ²˙1 = r˙0 = ˙ θ˜ =

−q0 ²1 + ρ∗ θ˜> φ − ρ˜r0 −(p0 + α0 (φ> Γφ)2 )r0 + φ> Γφ²1 sgn(ρ∗ ) −Γ²1 φ sgn(ρ ), ρ˜˙ = γ²1 r0 ∗

(6.4.49)

368

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL We propose the Lyapunov-like function V =

²21 Γ−1 ˜ ρ˜2 r2 θ+ + |ρ∗ |θ˜> + γ0 0 2 2 2γ 2

where γ0 > 0 is a constant to be selected. The time derivative of V along the trajectories of (6.4.49) is given by V˙

= −q0 ²21 − γ0 p0 r02 − γ0 α0 r02 (φ> Γφ)2 + γ0 ²1 r0 φ> Γφ sgn(ρ∗ ) ≤ −q0 ²21 − γ0 p0 r02 − γ0 α0 r02 (φ> Γφ)2 + γ0 |²1 | |r0 |φ> Γφ

By completing the squares we obtain · ¸2 q0 ²21 |r0 |φ> Γφ r2 (φ> Γφ)2 ˙ V ≤ −q0 − |²1 | − γ0 + γ02 0 − γ0 p0 r02 − γ0 α0 r02 (φ> Γφ)2 2 2 q0 2q0 · ¸ γ0 ²2 ≤ −q0 1 − γ0 p0 r02 − α0 − γ0 r02 (φ> Γφ)2 2 2q0 Because γ0 > 0 is arbitrary, used for analysis only, for any given α0 and q0 > 0, we can choose it as γ0 = 2α0 q0 leading to 2

² V˙ ≤ −q0 1 − γ0 p0 r02 ≤ 0 2 Hence, ²1 , r0 , ρ, θ ∈ L∞ and ²1 , r0 ∈ L2 . Because ρ, r0 ∈ L∞ , it follows from (6.4.41) that eˆ1 ∈ L∞ which implies that e1 = ²1 + eˆ1 ∈ L∞ . Hence, yp ∈ L∞ , which, together with   (sI − F )−1 gG−1 p (s)yp   1 (sI − F )−1 gyp   φ= (6.4.50)   yp (s + p0 )(s + p1 ) r ˙ ρ˙ ∈ implies that φ ∈ L∞ . Using ²1 , φ ∈ L∞ and ²1 ∈ L2 in (6.4.49) we have θ, T L∞ L2 . From the error equation (6.4.49) it follows that ²˙1 and, therefore, y˙ p ∈ L∞ which imply that φ˙ and φ1 ∈ L∞ . The second derivative e¨1 can be shown to be ˙ φ, ˙ r˙0 ∈ L∞ in (6.4.41). Because e¨1 = y¨p − s2 Wm (s)r and bounded by using θ, s2 Wm (s) is proper, it follows that y¨p ∈ L∞ , which, together with (6.4.50), implies that φ¨ ∈ L∞ . Because ω = (s + p0 )(s + p1 )φ, we have that ω ∈ L∞ and therefore T up and all signals are bounded. Because r0 ∈ L∞ L2 and ρ ∈ L∞ , it follows T T from (6.4.41) that eˆ1 ∈ L∞ L2 , which, together with ²1 ∈ L∞ L2 , implies that T e1 ∈ L∞ L2 . Because e˙ 1 ∈ L∞ and e1 ∈ L2 , it follows that e1 (t) → 0 as t → ∞ and the proof of (i) is complete. From r0 ∈ L2 , r˙0 ∈ L∞ we also have r0 (t) → 0 as t → ∞.

6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 369 (ii) First, we show that φ is PE if r is sufficiently rich of order 2n. Using the expression (6.4.50) for φ and substituting yp = Wm r + e1 , we can write φ = φm + φ¯ where



φm and

 (sI − F )−1 gG−1 p Wm  (sI − F )−1 gWm  1 r  =  Wm (s + p0 )(s + p1 )  1 

 (sI − F )−1 gG−1 p  (sI − F )−1 g  1   e1 φ¯ =  1 (s + p0 )(s + p1 )  0

Using the same arguments as in the proof of Theorem 6.4.2, we can establish that φm is PE provided r is sufficiently rich of order 2n and Zp , Rp are coprime. Then the PE property of φ follows immediately from Lemma 4.8.3 and e1 ∈ L2 . If kp is known, then ρ˜ = 0 and (6.4.49) is reduced to −q0 ²1 + ρ∗ θ˜> φ

²˙1 = ˙ θ˜ =

−Γ²1 φsgn(ρ∗ )

(6.4.51)

We can use the same steps as in the proof of Corollary 4.3.1 in Chapter 4 to show that the equilibrium ²1e = 0, θ˜e = 0 of (6.4.51) is e.s. provided φ is PE and ˙ φ ∈ L∞ , (ii) follows. φ, φ˙ ∈ L∞ . Since we have established in (i) that φ, (iii) When kp is unknown, we have ²˙1 = ˙ θ˜ =

−q0 ²1 + ρ∗ θ˜> φ − ρ˜r0 −Γ²1 φsgn(ρ∗ )

(6.4.52)

We consider (6.4.52) as a linear-time-varying system with ²1 , θ˜ as states and ρ˜r0 as the external input. As shown in (ii), when ρ˜r0 = 0 and φ is PE, T the homogeneous part of (6.4.52) is e.s. We have shown in (i) that ρ˜ ∈ L∞ , r0 ∈ L∞ L2 and r0 (t) → 0 as t → ∞. (by extending the results of Corollary 3.3.1) that T Therefore, it follows ˜ → 0 as t → ∞. ²1 , θ˜ ∈ L∞ L2 that ²1 (t), θ(t) (iv) Because ²1 , r0 ∈ L2 we have Z

t 0

Z |ρ˜˙ |dτ

t

≤ γ

|²1 ||r0 |dτ Z0 ∞ Z 1 2 2 ≤ γ( ²1 dτ ) ( 0

0

∞

1

r02 dτ ) 2 < ∞

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

which for t → ∞ implies that ρ˙ = ρ˜˙ ∈ L1 and therefore ρ, ρ˜ converge to a constant as t → ∞. 2

Example 6.4.3 Let us consider the third order plant yp =

s3

+ a2

kp up + a1 s + a0

s2

where kp , a0 , a1 , a2 are unknown constants, and the sign of kp is assumed to be known. The control objective is to choose up to stabilize the plant and force the output yp to track the output ym of the reference model given by ym =

1 r (s + 2)3

Because n∗ = 3, the MRAC scheme in Table 6.3 is considered. We choose p1 = p0 = 1 is SPR. The signals ω, φ, φ1 are generated as 2 so that Wm (s)(s + p1 )(s + p0 ) = s+2 · ¸ · ¸ · ¸ −10 −25 1 0 ω˙ 1 = ω1 + up , ω1 (0) = 1 0 0 0 · ¸ · ¸ · ¸ −10 −25 1 0 ω˙ 2 = ω2 + yp , ω2 (0) = 1 0 0 0 ω = [ω1> , ω2> , yp , r]> 1 1 φ1 = ω, φ = φ1 s+2 s+2 by choosing Λ(s) = (s + 5)2 . Then, according to Table 6.3, the adaptive control law that achieves the control objective is given by ˙ 0 +α2 (φ>Γφ)4 r0 −α0 (φ>Γφ)3 ²1 sgn(kp ) up = θ>ω−²1 sgn(kp )φ>Γφ1 −4α0 φ>Γφ(φ>Γφ)r 0 θ˙ = −Γ²1 φsgn(kp ), ρ˙ = γ²1 r0 ²1 = e1 − eˆ1 ,

e1 = yp − ym ,

eˆ1 =

1 ρr0 s+2

and r0 is generated by the equation r˙0 = −(2 + α0 (φ> Γφ)2 )r0 + φ> Γφ²1 sgn(kp ) The simulation results of the MRAC scheme for a unit step reference input are shown in Figure 6.12. The plant used for simulations is an unstable one with transfer function Gp (s) = s3 +6s21+3s−10 . The initial conditions for the controller parameters are: θ0 = [1.2, −9, 31, 160, −50, 9]> , and ρ(0) = 0.2. The design parameters used for simulations are: Γ = 50I, γ = 50, α0 = 0.01. 5

6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 371 0.1

0

-0.1

tracking error

-0.2

-0.3

-0.4

-0.5

-0.6

-0.7 0

5

10

15

20

25 sec

30

35

40

45

50

Figure 6.12 Response of the MRAC scheme for Example 6.4.3 with r(t)= unit step function.

Remark 6.4.6 The effect of initial conditions can be taken into account by using e1 = Wm (s)ρ∗ (up − θ∗> ω) + Cc> (sI − Ac )−1 e(0) instead of (6.4.33). The proof can be easily modified to take care of the exponentially decaying to zero term that is due to e(0) 6= 0 (see Problem 6.19). Similarly, the results presented here are valid only if the existence and uniqueness of solutions of (6.4.49) can be established. For further discussion and details on the existence and uniqueness of solutions of the class of differential equations that arise in adaptive systems, the reader is referred to [191].

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

Remark 6.4.7 The procedure for n∗ = 3 may be extended to the case of n∗ > 3 by following similar steps. The complexity of the control input up , however, increases considerably with n∗ to the point that it defeats any simplicity we may gain from analysis by using a single Lyapunovlike function to establish stability. In addition to complexity the highly nonlinear terms in the control law may lead to a “high bandwidth” control input that may have adverse effects on robustness with respect to modeling errors. We will address some of these robustness issues in Chapter 8. On the other hand, the idea of unnormalized adaptive laws together with the nonlinear modification of the certainty equivalence control laws were found to be helpful in solving the adaptive control problem for a class of nonlinear plants [98, 99, 105].

6.5

Direct MRAC Laws

with Normalized

Adaptive

In this section we present and analyze a class of MRAC schemes that dominated the literature of adaptive control due to the simplicity of their design as well as their robustness properties in the presence of modeling errors. Their design is based on the certainty equivalence approach that combines a control law, motivated from the known parameter case, with an adaptive law generated using the techniques of Chapter 4. The adaptive law is driven by the normalized estimation error and is based on an appropriate parameterization of the plant that involves the unknown desired controller parameters. While the design of normalized MRAC schemes follows directly from the results of Section 6.3 and Chapter 4, their analysis is more complicated than that of the unnormalized MRAC schemes presented in Section 6.4 for the case of n∗ = 1, 2. However, their analysis, once understood, carries over to all relative degrees of the plant without additional complications.

6.5.1

Example: Adaptive Regulation

Let us consider the scalar plant x˙ = ax + u,

x(0) = x0

(6.5.1)

6.5. DIRECT MRAC WITH NORMALIZED ADAPTIVE LAWS

373

where a is an unknown constant and −am is the desired closed-loop pole for some am > 0. The desired control law u = −k ∗ x,

k ∗ = a + am

that could be used to meet the control objective when a is known, is replaced with u = −k(t)x (6.5.2) where k(t) is to be updated by an appropriate adaptive law. In Section 6.2.1 we updated k(t) using an unnormalized adaptive law driven by the estimation error, which was shown to be equal to the regulation error x. In this section, we use normalized adaptive laws to update k(t). These are adaptive laws driven by the normalized estimation error which is not directly related to the regulation error x. As a result, the stability analysis of the closed-loop adaptive system is more complicated. As shown in Section 6.2.1, by adding and subtracting the term −k ∗ x in the plant equation (6.5.1) and using k ∗ = a + am to eliminate the unknown a, we can obtain the parametric plant model x˙ = −am x + k ∗ x + u whose transfer function form is x=

1 (k ∗ x + u) s + am

(6.5.3)

If we now put (6.5.3) into the form of the general parametric model z = W (s)θ∗> ψ considered in Chapter 4, we can simply pick any adaptive law for estimating k ∗ on-line from Tables 4.1 to 4.3 of Chapter 4. Therefore, let us rewrite (6.5.3) as 1 z= k∗ x (6.5.4) s + am 1 where z = x − s+a u is available from measurement. m Using Table 4.1 of Chapter 4, the SPR-Lyapunov design approach gives the adaptive law

k˙ = γ²x

374

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL ² = z − zˆ − zˆ =

1 ²n2 s + am s

1 kx, s + am

(6.5.5)

n2s = x2

where γ > 0 is the adaptive gain and L(s) in Table 4.1 is taken as L(s) = 1. 1 x, we use Table 4.2(A) to obtain Rewriting (6.5.4) as z = k ∗ φ, φ = s+a m the gradient algorithm k˙ = γ²φ z − zˆ , m2 = 1 + φ2 ² = m2 1 φ = x, zˆ = kφ, γ > 0 s + am

(6.5.6)

and from Table 4.3(A), the least-squares algorithm 2 2

p φ k˙ = p²φ, p˙ = − 2 , p(0) > 0 m 1 z − zˆ , zˆ = kφ, m2 = 1 + φ2 , φ = x ² = 2 m s + am

(6.5.7)

The control law (6.5.2) with any one of the three adaptive laws (6.5.5) to (6.5.7) forms an adaptive control scheme. We analyze the stability properties of each scheme when applied to the plant (6.5.1) as follows: We start by writing the closed-loop plant equation as 1 ˜ x= (−kx) (6.5.8) s + am by substituting u = −kx in (6.5.3). As shown in Chapter 4 all three adaptive laws guarantee that k˜ ∈ L∞ independent of the boundedness of x, u, which implies from (6.5.8) that x cannot grow or decay faster than an exponential. However, the boundedness of k˜ by itself does not imply that x ∈ L∞ , let alone x(t) → 0 as t → ∞. To analyze (6.5.8), we need to exploit the ˜ by using the properties of the specific adaptive law that properties of kx generates k(t). Let us start with the adaptive law (6.5.5). As shown in Chapter 4 using the Lyapunov-like function V =

²2 k˜2 + 2 2γ

6.5. DIRECT MRAC WITH NORMALIZED ADAPTIVE LAWS

375

and its time derivative V˙ = −am ²2 − ²2 n2s ≤ 0 ˙ the adaptive law (6.5.5) guarantees that ², k˜ ∈ L∞ and ², ²ns , k˜ ∈ L2 independent of the boundedness of x. The normalized estimation error ² is ˜ through the equation related to kx ² = z − zˆ −

1 1 ˜ − ²n2 ) ²n2 = (−kx s s + am s s + am

(6.5.9)

where n2s = x2 . Using (6.5.9) and ²n2s = ²ns x in (6.5.8), we obtain x=²+ T

1 1 ²n2s = ² + ²ns x s + am s + am

(6.5.10)

Because ² ∈ L∞ L2 and ²ns ∈ L2 the boundedness of x is established by taking absolute values on each side of (6.5.10) and applying the B-G lemma. We leave this approach as an exercise for the reader. A more elaborate but yet more systematic method that we will follow in the higher order case involves the use of the properties of the L2δ norm and the B-G Lemma. We present such a method below and use it to understand the higher-order case to be considered in the sections to follow. Step 1. Express the plant output y (or state x) and plant input u in ˜ We have terms of the parameter error k. x=

1 ˜ (−kx), s + am

u = (s − a)x =

(s − a) ˜ (−kx) s + am

(6.5.11)

The above integral equations may be expressed in the form of algebraic inequalities by using the properties of the L2δ norm k(·)t k2δ , which for simplicity we denote by k · k. We have ˜ ˜ kxk ≤ ckkxk, kuk ≤ ckkxk (6.5.12) where c ≥ 0 is a generic symbol used to denote any finite constant. Let us now define 4 m2f = 1 + kxk2 + kuk2 (6.5.13) The significance of the signal mf is that it bounds |x|, |x| ˙ and |u| from above provided k ∈ L∞ . Therefore if we establish that mf ∈ L∞ then the boundedness of all signals follows. The boundedness of |x|/mf , |x|/m ˙ f , |u|/mf follows

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

from k˜ ∈ L∞ and the properties of the L2δ -norm given by Lemma 3.3.2, i.e., from (6.5.11) we have °

°

|x(t)| ° 1 ° ˜ kxk ≤ c ° |k| ≤° ° mf s + am °2δ mf and

|x(t)| ˙ |x(t)| ˜ |x(t)| ≤ c ≤ am + |k| mf mf mf

Similarly, |x| |u(t)| ≤ |k| ≤c mf mf Because of the normalizing properties of mf , we refer to it as the fictitious normalizing signal. It follows from (6.5.12), (6.5.13) that ˜ 2 m2f ≤ 1 + ckkxk

(6.5.14)

Step 2. Use the Swapping Lemma and properties of the L2δ norm to ˜ with terms that are guaranteed by the adaptive law to have upper bound kkxk finite L2 gains. We use the Swapping Lemma A.2 given in Appendix A to write the identity µ

˜ = 1− kx

¶

α0 1 α0 ˜ ˜ = ˜˙ + k˜x) ˜ + α0 kx (kx ˙ + kx kx s + α0 s + α0 s + α0 s + α0

˜ = −(s+am )x, where α0 > 0 is an arbitrary constant. Since, from (6.5.11), kx we have (s + am ) 1 ˜˙ + k˜x) ˜ = ˙ − α0 kx (kx x (6.5.15) s + α0 (s + α0 ) which imply that ° °

˜ ≤° kkxk °

°

°

°

° s + am ° 1 ° ˜˙ + kk˜xk) ° (kkxk ° ˙ + α0 ° ° ° s + α ° kxk s + α0 ∞δ 0 ∞δ

1 For α0 > 2am > δ, we have k s+α k∞δ = 0

˜ ≤ kkxk

2 2α0 −δ

<

2 α0 ,

therefore,

2 ˜˙ (kkxk + kk˜xk) ˙ + α0 ckxk α0

6.5. DIRECT MRAC WITH NORMALIZED ADAPTIVE LAWS m where c = k s+a s+α0 k∞δ . Since

˜ ≤ kkxk

x x˙ mf , mf

377

∈ L∞ , it follows that

c ˜˙ ˜ f k) + α0 ckxk (kkmf k + kkm α0

(6.5.16)

Equation (6.5.16) is independent of the adaptive law used to update k(t). ˜˙ f k in (6.5.16) is “small” because k˙ ∈ L2 (guaranteed by any The term αc0 kkm ˜ f k can one of the adaptive laws (6.5.5) - (6.5.7)), whereas the term c kkm α0

be made small by choosing α0 large but finite. Large α0 , however, may make α0 ckxk large unless kxk is also small in some sense. We establish the smallness of the regulation error x by exploiting its relationship with the normalized estimation error ². This relationship depends on the specific adaptive law used. For example, for the adaptive law (6.5.5) that is based on the SPR-Lyapunov design approach, we have established that x=²+

1 ²n2 s + am s

|x| which together with |²n2s | ≤ |²ns | m mf ≤ c²ns mf imply that f

kxk ≤ k²k + ck²ns mf k hence, ˜ ≤ kkxk

c ˜˙ ˜ f k) + α0 ck²k + α0 ck²ns mf k (kkmf k + kkm α0

(6.5.17)

Similarly, for the gradient or least-squares algorithms, we have x = ²m2 +

1 ˙ kφ s + am

(6.5.18)

obtained by using the equation 1 1 ˙ kx = kφ − kφ s + am s + am that follows from Swapping Lemma A.1 together with the equation for ²m2 in (6.5.6). Equation (6.5.18) implies that ˙ kxk ≤ k²k + k²n2s k + ckkφk

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

Because n2s = φ2 and φ = φ mf ∈ L∞ and, therefore,

1 s+am x,

we have |φ(t)| ≤ ckxk which implies that

˙ fk kxk ≤ k²k + k²ns mf k + ckkm ˜ as Substituting for kxk in (6.5.16), we obtain the same expression for kkxk in (6.5.17). Step 3. Use the B-G Lemma to establish boundedness. From (6.5.14) and (6.5.17), we obtain m2f ≤ 1 + α02 c +

c ˜˙ ˜ f k2 ) + cα2 k²ns mf k2 (kkmf k2 + kkm 0 α02

by using the fact that ² ∈ L∞

T

(6.5.19)

L2 . We can express (6.5.19) as

m2f ≤ 1 + α02 c +

c kmf k2 + cα02 k˜ g mf k2 α02

(6.5.20)

˜˙ 2 4 ˙ where g˜2 = |²ns |2 + |αk|4 . Because the adaptive laws guarantee that ²ns , k˜ ∈ 0 L2 it follows that g˜ ∈ L2 . Using the definition of the L2δ norm, inequality (6.5.20) may be rewritten as

m2f

≤1+

cα02

+c

Z t 0

µ

e

−δ(t−τ )

¶

α02 g˜2 (τ )

1 + 2 m2f (τ )dτ α0

Applying the B-G Lemma III, we obtain m2f ≤ (1 + cα02 )e−δ(t−τ ) Φ(t, t0 ) + (1 + cα02 )δ where

c 2 (t−τ )

Φ(t, τ ) = e α0

ec

Rt τ

Z t t0

e−δ(t−τ ) Φ(t, τ )dτ

α20 g˜2 (σ)dσ

Choosing α0 so that αc2 ≤ 2δ , α0 > 2am and using g˜ ∈ L2 , it follows that 0 mf ∈ L∞ . Because mf bounds x, x, ˙ u from above, it follows that all signals in the closed-loop adaptive system are bounded. Step 4. Establish convergence of the regulation error to zero. For the adaptive law (6.5.5), it follows from (6.5.9), (6.5.10) that x ∈ L2 and from (6.5.8) that x˙ ∈ L∞ . Hence, using Lemma 3.2.5, we have x(t) → 0 as t → ∞. For the adaptive law (6.5.6) or (6.5.7) we have from (6.5.18) that x ∈ L2 and from (6.5.8) that x˙ ∈ L∞ , hence, x(t) → 0 as t → ∞.

6.5. DIRECT MRAC WITH NORMALIZED ADAPTIVE LAWS

6.5.2

379

Example: Adaptive Tracking

Let us consider the tracking problem defined in Section 6.2.2 for the first order plant x˙ = ax + bu (6.5.21) where a, b are unknown (with b 6= 0). The control law u = −k ∗ x + l∗ r

(6.5.22)

where

am + a bm , l∗ = (6.5.23) b b guarantees that all signals in the closed-loop plant are bounded and the plant state x converges exponentially to the state xm of the reference model k∗ =

xm =

bm r s + am

(6.5.24)

Because a, b are unknown, we replace (6.5.22) with u = −k(t)x + l(t)r

(6.5.25)

where k(t), l(t) are the on-line estimates of k ∗ , l∗ , respectively. We design the adaptive laws for updating k(t), l(t) by first developing appropriate parametric models for k ∗ , l∗ of the form studied in Chapter 4. We then choose the adaptive laws from Tables 4.1 to 4.5 of Chapter 4 based on the parametric model satisfied by k ∗ , l∗ . As in Section 6.2.2, if we add and subtract the desired input −bk ∗ x+bl∗ r in the plant equation (6.5.21) and use (6.5.23) to eliminate the unknown a, we obtain x˙ = −am x + bm r + b(u + k ∗ x − l∗ r) which together with (6.5.24) and the definition of e1 = x − xm give e1 =

b (u + k ∗ x − l∗ r) s + am

(6.5.26)

Equation (6.5.26) can also be rewritten as e1 = b(θ∗> φ + uf )

(6.5.27)

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

1 1 where θ∗ = [k ∗ , l∗ ], φ = s+a [x, −r]> , uf = s+a u. Both equations are in m m the form of the parametric models given in Table 4.4 of Chapter 4. We can use them to choose any adaptive law from Table 4.4. As an example, let us choose the gradient algorithm listed in Table 4.4(D) that does not require the knowledge of sign b. We have

k˙ = N (w)γ1 ²φ1 l˙ = N (w)γ2 ²φ2 ˆb˙ = N (w)γ²ξ N (w) = w2 cos w, w = w0 +

ˆb2 2γ

w˙ 0 = ²2 m2 , w0 (0) = 0 e1 − eˆ1 , eˆ1 = N (w)ˆbξ ² = m2 ξ = kφ1 + lφ2 + uf , uf =

(6.5.28) 1 u s + am

1 1 x, φ2 = − r s + am s + am = 1 + n2s , n2s = φ21 + φ22 + u2f

φ1 = m2

γ1 , γ2 , γ > 0 As shown in Chapter 4, the above adaptive law guarantees that k, l, w, ˙ l,˙ ˆb˙ ∈ L∞ T L2 independent of the boundedness of w0 ∈ L∞ and ², ²ns , k, u, e1 , φ. Despite the complexity of the adaptive law (6.5.28), the stability analysis of the closed-loop adaptive system described by the equations (6.5.21), (6.5.25), (6.5.28) is not more complicated than that of any other adaptive law from Table 4.4. We carry out the stability proof by using the properties of the L2δ -norm and B-G Lemma in a similar way as in Section 6.5.1. Step 1. Express the plant output x and input u in terms of the parameter ˜ ˜l. From (6.5.24), (6.5.25) and (6.5.26) we have errors k, x = xm −

³ ´ ³ ´ b 1 ˜ − ˜lr = ˜ kx bm r + b˜lr − bkx s + am s + am

(6.5.29)

6.5. DIRECT MRAC WITH NORMALIZED ADAPTIVE LAWS

381

and from (6.5.21), (6.5.29) u=

i (s − a) (s − a) h ˜ x= bm r + b˜lr − bkx b b(s + am )

(6.5.30)

For simplicity, let us denote k(·)t k2δ by k · k. Again for the sake of clarity and ease of exposition, let us also denote any positive finite constant whose actual value does not affect stability with the same symbol c. Using the properties of the L2δ -norm in (6.5.29), (6.5.30) and the fact that r, ˜l ∈ L∞ we have ˜ ˜ kxk ≤ c + ckkxk, kuk ≤ c + ckkxk for any δ ∈ [0, 2am ), which imply that the fictitious normalizing signal defined as 4 m2f = 1 + kxk2 + kuk2 satisfies ˜ 2 m2f ≤ c + ckkxk

(6.5.31)

˜ that φ1 /mf , x/m We verify, using the boundedness of r, ˜l, k, ˙ f , ns /mf ∈ L∞ as follows: From the definition of φ1 , we have |φ1 (t)| ≤ ckxk ≤ cmf . Simi˜ we have larly, from (6.5.29) and the boundedness of r, ˜l, k, |x(t)| ≤ c + ckxk ≤ c + cmf ˜ it follows that |x| Because x˙ = −am x + bm r + b˜lr − bkx, ˙ ≤ c + cmf . Next, 2 let us consider the signal ns = 1 + φ21 + φ22 + u2f . Because |uf | ≤ ckuk ≤ cmf φ1 , φ2 ∈ L∞ , it follows that ns ≤ cmf . and m f Step 2. Use the Swapping Lemma and properties of the L2δ norm to ˜ with terms that are guaranteed by the adaptive law to have upper bound kkxk finite L2 -gains. We start with the identity µ

¶

µ

¶

α0 1 ˜ = ˜˙ + k˜x˙ + α0 kx ˜ ˜ + α0 kx kx kx s + α0 s + α0 s + α0 s + α0 (6.5.32) where α0 > 0 is an arbitrary constant. From (6.5.29) we also have that ˜ = 1− kx

˜ = − (s + am ) e1 + ˜lr kx b

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

where e1 = x − xm , which we substitute in the second term of the right-hand side of (6.5.32) to obtain µ

˜ = kx

¶

1 ˜˙ + k˜x˙ − α0 (s + am ) e1 + α0 ˜lr kx s + α0 b (s + α0 ) s + α0

˜ ˜l, r ∈ L∞ we have Because k, ˜ ≤ kkxk

c ˜˙ c kkxk + kxk ˙ + α0 cke1 k + c α0 α0

(6.5.33)

for any 0 < δ < 2am < α0 . As in Section 6.5.1, the gain of the first two terms on the right-hand side of (6.5.33) can be reduced by choosing α0 large. So the only term that needs further examination is α0 cke1 k. The tracking error e1 , however, is related to the normalized estimation error ² through the equation e1 = ²m2 + N (w)ˆbξ = ² + ²n2s + N (w)ˆbξ that follows from (6.5.28). T Because ², ²ns ∈ L∞ L2 and N (w)ˆb ∈ L∞ , the signal we need to concentrate on is ξ which is given by ξ = kφ1 + lφ2 +

1 u s + am

We consider the equation ³ ´ 1 1 1 ˙ 1 + lr ˙ u= (−kx + lr) = −kφ1 − lφ2 + kφ s + am s + am s + am

obtained by using the Swapping Lemma A.1 or the equation ˙ 1 + lφ ˙ 2 ) = u + kφ ˙ 1 + lφ ˙ 2 (s + am )(kφ1 + lφ2 ) = kx − lr + (kφ Using any one of the above equations to substitute for k1 φ1 + lφ2 in the equation for ξ we obtain ξ=

1 ˙ 1 + lφ ˙ 2) (kφ s + am

hence e1 = ² + ²n2s + N (w)ˆb

1 ˙ 1 + lφ ˙ 2) (kφ s + am

(6.5.34)

6.5. DIRECT MRAC WITH NORMALIZED ADAPTIVE LAWS

383

Because ², N (w), ˆb, l,˙ φ2 , r ∈ L∞ it follows from (6.5.34) that ˙ 1k ke1 k ≤ c + k²n2s k + ckkφ

(6.5.35)

and, therefore, (6.5.33) and (6.5.35) imply that ˜ ≤ c + cα0 + kkxk

c ˜˙ c ˙ 1k kkxk + kxk ˙ + α0 ck²n2s k + α0 ckkφ α0 α0

(6.5.36)

Step 3. Use the B-G Lemma to establish boundedness. Using (6.5.36) and the normalizing properties of mf , we can write (6.5.31) in the form m2f

c ˙ c ˙ f k2 kkmf k2 + 2 kmf k2 + α02 ck²ns mf k2 + α02 ckkm 2 α0 α0 c ≤ c + cα02 + cα02 k˜ g mf k2 + 2 kmf k2 (6.5.37) α0 ≤ c + cα02 +

4 ˙ 2 + |²2 n2 | + |k|. ˙ Inequality (6.5.37) has exactly the same where g˜2 = α14 |k| s 0 form and properties as inequality (6.5.20) in Section 6.5.1. Therefore the boundedness of mf follows by applying the B-G Lemma and choosing α02 ≥ max{4a2m , 2c δ } as in the example of Section 6.5.1. From mf ∈ L∞ we have x, x, ˙ ns , φ1 ∈ L∞ , which imply that u and all signals in the closed-loop plant are bounded.

Step 4. Establish convergence of the tracking error to zero. We show the ˙ l˙ ∈ convergence of the tracking error to zero by using (6.5.34). From ², ²ns , k, ˆ L2 and ns , N (w)b, φ1 , φ2 ∈ L∞ we can establish, using (6.5.34), that e1 ∈ L2 which together with e˙ 1 = x˙ − x˙ m ∈ L∞ imply (see Lemma 3.2.5) that e1 (t) → 0 as t → ∞.

6.5.3

MRAC for SISO Plants

In this section we extend the design approach and analysis used in the examples of Sections 6.5.1 and 6.5.2 to the general SISO plant (6.3.1). We consider the same control objective as in Section 6.3.1 where the plant (6.3.1) and reference model (6.3.4) satisfy assumptions P1 to P4, and M1 and M2, respectively. The design of MRAC schemes for the plant (6.3.1) with unknown parameters is based on the certainty equivalence approach and is conceptually simple. With this approach, we develop a wide class of MRAC schemes by

384

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

combining the MRC law (6.3.22), where θ∗ is replaced by its estimate θ(t), with different adaptive laws for generating θ(t) on-line. We design the adaptive laws by first developing appropriate parametric models for θ∗ which we then use to pick up the adaptive law of our choice from Tables 4.1 to 4.5 in Chapter 4. Let us start with the control law up = θ1> (t)

α(s) α(s) up + θ2> (t) yp + θ3 (t)yp + c0 (t)r Λ(s) Λ(s)

(6.5.38)

whose state-space realization is given by ω˙ 1 = F ω1 + gup ,

ω1 (0) = 0

ω˙ 2 = F ω2 + gyp ,

ω2 (0) = 0

(6.5.39)

up = θ> ω where θ = [θ1> , θ2> , θ3 , c0 ]> and ω = [ω1> , ω2> , yp , r]> , and search for an adaptive law to generate θ(t), the estimate of the desired parameter vector θ∗ . In Section 6.4.1 we develop the bilinear parametric model e1 = Wm (s)ρ∗ [up − θ∗> ω]

(6.5.40)

1 ∗ ∗> ∗> ∗ ∗ > c∗0 , θ = [θ1 , θ2 , θ3 , c0 ] by adding and subtracting the desired input θ∗> ω in the overall representation of the plant and controller

where ρ∗ =

control states (see (6.4.6)). The same parametric model may be developed by using the matching equation (6.3.13) to substitute for the unknown plant polynomial Rp (s) in the plant equation and by cancelling the Hurwitz polynomial Zp (s). The parametric model (6.5.40) holds for any relative degree of the plant transfer function. A linear parametric model for θ∗ may be developed from (6.5.40) as follows: Because ρ∗ = c1∗ and θ∗> ω = θ0∗> ω0 + c∗0 r where θ0∗ = [θ1∗> , θ2∗> , θ3∗ ]> 0

and ω0 = [ω1> , ω2> , yp ]> , we rewrite (6.5.40) as Wm (s)up = c∗0 e1 + Wm (s)θ0∗> ω0 + c∗0 Wm (s)r Substituting for e1 = yp − ym and using ym = Wm (s)r we obtain Wm (s)up = c∗0 yp + Wm (s)θ0∗> ω0

6.5. DIRECT MRAC WITH NORMALIZED ADAPTIVE LAWS

385

which may be written as z = θ∗> φp

(6.5.41)

where z = Wm (s)up φp = [Wm (s)ω1> , Wm (s)ω2> , Wm (s)yp , yp ]> θ∗ = [θ1∗> , θ2∗> , θ3∗ , c∗0 ]> In view of (6.5.40), (6.5.41) we can now develop a wide class of MRAC schemes by using Tables 4.1 to 4.5 to choose an adaptive law for θ based on the bilinear parametric model (6.5.40) or the linear one (6.5.41). Before we do that, let us compare the two parametric models (6.5.40), (6.5.41). The adaptive laws based on (6.5.40) listed in Table 4.4 of Chapter 4 generate estimates for c∗0 as well as for ρ∗ = c1∗ . In addition, some 0 algorithms require the knowledge of the sgn(ρ∗ ) and of a lower bound for |ρ∗ |. On the other hand the adaptive laws based on the linear model (6.5.41) generate estimates of c∗0 only, without any knowledge of the sgn(ρ∗ ) or lower bound for |ρ∗ |. This suggests that (6.5.41) is a more appropriate parameterization of the plant than (6.5.40). It turns out, however, that in the stability analysis of the MRAC schemes whose adaptive laws are based on (6.5.41), 1/c0 (t) is required to be bounded. This can be guaranteed by modifying the adaptive laws for c0 (t) using projection so that |c0 (t)| ≥ c0 > 0, ∀t ≥ 0 for some constant c0 ≤ |c∗0 | = | kkmp |. Such a projection algorithm requires ³

´

the knowledge of the sgn(c∗0 ) = sgn kkmp and the lower bound c0 , which is calculated from the knowledge of an upper bound for |kp |. Consequently, as far as a priori knowledge is concerned, (6.5.41) does not provide any special advantages over (6.5.40). In the following we modify all the adaptive laws that are based on (6.5.41) using the gradient projection method so that |c0 (t)| ≥ c0 > 0 ∀t ≥ 0. The main equations of several MRAC schemes formed by combining (6.5.39) with an adaptive law from Tables 4.1 to 4.5 based on (6.5.40) or (6.5.41) are listed in Tables 6.4 to 6.7. The basic block diagram of the MRAC schemes described in Table 6.4 is shown in Figure 6.13. An equivalent representation that is useful for analysis is shown in Figure 6.14.

386

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL Table 6.4 MRAC schemes Z (s)

Plant

yp = kp Rpp (s) up

Reference model

Zm (s) ym = km R r = Wm (s)r m (s)

Control law

ω˙ 1 = F ω1 + gup , ω1 (0) = 0 ω˙ 2 = F ω2 + gyp , ω2 (0) = 0 up = θ> ω θ = [θ1> , θ2> , θ3 , c0 ]> , ω = [ω1> , ω2> , yp , r]> ωi ∈ Rn−1 , i = 1, 2

Adaptive law

Any adaptive law from Tables 6.5, 6.6

Assumptions

Plant and reference model satisfy assumptions P1 to P4 and M1, M2 respectively.

Design variables

α(s) F, g chosen so that (sI − F )−1 g = Λ(s) , where n−2 n−3 > α(s) = [s ,s , . . . , s, 1] for n ≥ 2 and α(s) = 0 for n = 1; Λ(s) = sn−1 + λn−2 sn−2 + · · · + λ0 is Hurwitz.

Figure 6.14 is obtained by rewriting up as up = θ∗> ω + θ˜> ω 4

where θ˜ = θ − θ∗ , and by using the results of Section 6.4.1, in particular, equation (6.4.6) to absorb the term θ∗> ω. For θ˜ = θ − θ∗ = 0, the closed-loop MRAC scheme shown in Figure 6.14 reverts to the one in the known parameter case shown in Figure 6.5. For θ˜ 6= 0, the stability of the closed-loop MRAC scheme depends very much on the properties of the input c1∗ θ˜> ω, which, in turn, depend on the properties 0 ˜ = θ(t) − θ∗ . of the adaptive law that generates the trajectory θ(t)

6.5. DIRECT MRAC WITH NORMALIZED ADAPTIVE LAWS

387

Table 6.5 Adaptive laws based on e1 = Wm (s)ρ∗ (up − θ∗> ω) A. Based on the SPR-Lyapunov approach Parametric model

Adaptive law

e1 = Wm (s)L(s)[ρ∗ (uf − θ∗> φ)] θ˙ = −Γ²φ sgn(kp /km ) ρ˙ = γ²ξ, ² = e1 − eˆ1 − Wm (s)L(s)(²n2s ) eˆ1 = Wm (s)L(s)[ρ(uf − θ> φ)] ξ = uf − θ> φ, φ = L−1 (s)ω uf = L−1 (s)up , n2s = φ> φ + u2f

Assumptions

Sign (kp ) is known

Design variables

Γ = Γ> > 0, γ > 0; Wm (s)L(s) is proper and SPR; L−1 (s) is proper and has stable poles B. Gradient algorithm with known sgn(kp )

Parametric model

Adaptive law

Design variables

e1 = ρ∗ (uf − θ∗> φ) θ˙ = −Γ²φ sgn(kp /km ) ρ˙ = γ²ξ e1 ² = e1m−ˆ 2 eˆ1 = ρ(uf − θ> φ) φ = Wm (s)ω, uf = Wm (s)up ξ = uf − θ> φ m2 = 1 + φ> φ + u2f

Γ = Γ> > 0, γ > 0

388

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL Table 6.5 (Continued) C. Gradient algorithm with unknown sgn(kp )

Parametric model

Adaptive law

Design variables

e1 = ρ∗ (uf − θ∗> φ) θ˙ = −N (x0 )Γ²φ ρ˙ = N (x0 )γ²ξ N (x0 ) = x20 cos x0 ρ2 x0 = w0 + 2γ , w˙ 0 = ²2 m2 , w0 (0) = 0 e1 −ˆ e1 ² = m2 , eˆ1 = N (x0 )ρ(uf − θ> φ) φ = Wm (s)ω, uf = Wm (s)up ξ = uf − θ> φ, m2 = 1 + n2s , n2s = φ> φ + u2f

Γ = Γ> > 0, γ > 0

The following theorem gives the stability properties of the MRAC scheme shown in Figures 6.13 and 14 when the adaptive laws in Tables 6.5 and 6.6 are used to update θ(t) on-line. Theorem 6.5.1 The closed-loop MRAC scheme shown in Figure 6.13 and described in Table 6.4 with any adaptive law from Tables 6.5 and 6.6 has the following properties: (i)

All signals are uniformly bounded.

(ii) The tracking error e1 = yp − ym converges to zero as t → ∞. (iii) If the reference input signal r is sufficiently rich of order 2n, r˙ ∈ L∞ and Rp , Zp are coprime, the tracking error e1 and parameter error θ˜ = θ − θ∗ converge to zero for the adaptive law with known sgn(kp ). The convergence is asymptotic in the case of the adaptive law of Table 6.5(A, B) and exponential in the case of the adaptive law of Table 6.6.

6.5. DIRECT MRAC WITH NORMALIZED ADAPTIVE LAWS

389

Table 6.6 Adaptive laws based on Wm (s)up = θ∗> φp g(θ) = c0 − c0 sgnc0 ≤ 0 θ = [θ1> , θ2> , θ3 , c0 ]>

Constraint

  Γx

Projection operator

if |c0 (t)| > c0 or if |c0 (t)| = c0 and (Γx)> ∇g ≤ 0 Pr [Γx]=  Γx−Γ ∇g∇g> Γx otherwise ∇g > Γ∇g 4

A. Gradient Algorithm Adaptive law

θ˙ = P r[Γ²φp ]

Design variable Γ = Γ> > 0 B. Integral gradient aAlgorithm

Adaptive law

Design variable

θ˙ = P r[−Γ(Rθ + Q)] φp φ> R˙ = −βR + m2p , R(0) = 0 φ Q˙ = −βQ − mp2 z, Q(0) = 0 z = Wm (s)up Γ = Γ> > 0, β > 0

C. Least-squares with covariance resetting Adaptive law

θ˙ = P r[P ²φp ]  >

−P φp φ P   m2 p if |c0 (t)| > c0 or

˙ P=

if |c0 (t)| = c0 and (P ²φp )> 5 g ≤ 0 0 otherwise + P (tr ) = P0 = ρ0 I P (0) = P > (0) > 0, tr is the time for which λmin (P (t)) ≤ ρ1 , ρ1 > ρ0 > 0  

Design variable

Common signals and variables W (s)u −ˆ z ² = m m2 p , zˆ = θ> φp , m2 = 1+φ> p φp > > φp = [Wm (s)ω1 , Wm (s)ω2 , Wm (s)yp , yp ]> |c0 (0)| ≥ c0 , 0 < c0 ≤ |c∗0 |, sgn(c0 (0)) = sgn(kp /km )

390

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL Table 6.7 Hybrid MRAC

Parametric model

Wm (s)up = θ∗> φp

ω˙ 1 = F ω1 + gup ω˙ 2 = F ω2 + gyp up (t) = θk> ω(t), t ∈ (tk , tk+1 ]

Control law

n

Hybrid adaptive law

o

R

t θk+1=P¯ r θk+Γ tkk+1²(τ )φp (τ )dτ ; k=0, 1, 2, . . . W (s)u −ˆ z

² = m m2 p , zˆ(t) = θk> φp , t ∈ (tk , tk+1 ] m2 = 1 + βφ> p φp where P¯ r[·] is the discrete-time projection ( x if |x0 | ≥ c0 4 P¯ r{x} = 0 x + γγ00 (c0 − x0 )sgn(c0 ) otherwise where x0 is the last element of the vector x; γ0 is the last column of the matrix Γ, and γ00 is the last element of γ0

Design variables

β > 1; tk = kTs , Ts > 0; Γ = Γ> > 0 2 − Ts λmax (Γ) > γ for some γ > 0

Outline of Proof The proof follows the same procedure as that for the examples presented in Sections 6.5.1, 6.5.2. It is completed in five steps. Step 1. Express the plant input and output in terms of the adaptation error θ˜> ω. Using Figure 6.14 we can verify that the transfer function between the input r + c1∗ θ˜> ω and the plant output yp is given by 0

µ ¶ 1 yp = Gc (s) r + ∗ θ˜> ω c0 where Gc (s) =

(1 −

θ1∗> (sI

−

c∗0 kp Zp − kp Zp [θ2∗> (sI − F )−1 g + θ3∗ ]

F )−1 g)Rp

Because of the matching equations (6.3.12) and (6.3.13), and the fact that (sI − α(s) F )−1 g = Λ(s) , we have, after cancellation of all the stable common zeros and poles

6.5. DIRECT MRAC WITH NORMALIZED ADAPTIVE LAWS

ym

- Wm (s)

r

− ?e 1 Σl+ yp 6

¡ µ + l up - Gp (s) Σ + + µ ¡£± 6 ¡ ? ? ¡ £+ −1 ¡ £ (sI − F ) g (sI − F )−1 g

- c0 ¡ ¡

391

θ1> ¤¡

¤¡

¡ µ ¾ ω1

¡

¤¡

θ2>

¡ µ ω2 ¾ ¡ µ ¾ θ3

¡ ¡

Adaptive Law ¾ from Tables 6.5 to 6.7

yp , ym , up

Figure 6.13 Block diagram of direct MRAC with normalized adaptive law. in Gc (s), that Gc (s) = Wm (s). Therefore, the plant output may be written as µ ¶ 1 ˜> yp = Wm (s) r + ∗ θ ω (6.5.42) c0 Because yp = Gp (s)up and G−1 p (s) has stable poles, we have ¶ µ 1 ˜> −1 up = Gp (s)Wm (s) r + ∗ θ ω c0

(6.5.43)

where G−1 p (s)Wm (s) is biproper Because of Assumption M2. We now define the fictitious normalizing signal mf as 4

m2f = 1 + kup k2 + kyp k2

(6.5.44)

where k · k denotes the L2δ -norm for some δ > 0. Using the properties of the L2δ norm it follows that mf ≤ c + ckθ˜> ωk (6.5.45)

392

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL 1 ˜> c∗0 θ ω

1 ¾ c∗0

˜> Adaptive ¾ × ¾θ Law

ω

6 + l up ω yp - Gp (s) Σ + + µ£± 6 ¡ ¡ ? ? £ ¡ + −1 £ ¡ (sI − F ) g (sI − F )−1 g

+ r - ? l - c∗ Σ 0 +

-

θ1∗> ¾ ω1 θ2∗> ¾

ω2 θ3∗ ¾

Figure 6.14 Equivalent representation of the MRAC scheme of Table 6.4. where c is used to denote any finite constant and δ > 0 is such that Wm (s − δ δ −1 2 ), Gp (s − 2 ) have stable poles. Furthermore, for θ ∈ L∞ (guaranteed by the adaptive law), the signal mf bounds most of the signals and their derivatives from above. Step 2. Use the Swapping Lemmas and properties of the L2δ norm to upper bound kθ˜> ωk with terms that are guaranteed by the adaptive law to have finite L2 gains. This is the most complicated step and it involves the use of Swapping Lemmas A.1 and A.2 to obtain the inequality ∗ c mf + cα0n k˜ g mf k α0

kθ˜> ωk ≤

(6.5.46)

4

˙ 2 + ²2 and g˜ is guaranteed by the adaptive law to belong to where g˜2 = ²2 n2s + |θ| L2 and α0 > 0 is an arbitrary constant to be chosen. Step 3. Use the B-G Lemma to establish boundedness. From (6.5.45) and (6.5.46), it follows that m2f ≤ c +

∗ c 2 m + cα02n k˜ g mf k2 α02 f

or

Z m2f

t

≤c+c 0

∗

α02n g˜2 (τ )m2f (τ )dτ

(6.5.47)

6.5. DIRECT MRAC WITH NORMALIZED ADAPTIVE LAWS

393

for any α0 > α0∗ and some α0∗ > 0. Applying the B-G Lemma and using g˜ ∈ L2 , the boundedness of mf follows. Using mf ∈ L∞ , we establish the boundedness of all the signals in the closed-loop plant. Step 4. Show that the tracking error converge to zero. The convergence of e1 to zero is established by showing that e1 ∈ L2 and e˙ 1 ∈ L∞ and using Lemma 3.2.5. Step 5. Establish that the parameter error converges to zero. The convergence of the estimated parameters to their true values is established by first showing that the signal vector φ or φp , which drives the adaptive law under consideration, can be expressed as φ or φp = H(s)r + φ¯ where H(s) is a stable transfer matrix and φ¯ ∈ L2 . If r is sufficiently rich of order 2n and Zp , Rp are coprime then it follows from the results of Section 6.4 that φm = H(s)r is PE, which implies that φ or φp is PE. The PE property of φp or φ guarantees that θ˜ and e1 converge to zero as shown in Chapter 4. A detailed proof of Theorem 6.5.1 is given in Section 6.8. 2

The MRAC scheme of Table 6.4 with any adaptive law from Table 6.5 guarantees that ξ ∈ L2 which together with ² ∈ L2 implies that ρ˙ ∈ L1 , i.e., Z t 0

|ρ|dτ ˙ ≤γ

µZ ∞ 0

2

² dτ

¶ 1 µZ ∞ 2 0

¶1 2

ξ dτ

2

<∞

which, in turn, implies that ρ(t) converges to a constant as t → ∞ independent of the richness of r. The stability properties of the hybrid MRAC scheme of Table 6.7 are similar to the continuous-time MRAC schemes and are summarized by the following Theorem. Theorem 6.5.2 The closed-loop system obtained by applying the hybrid MRAC scheme of Table 6.7 to the plant given in Table 6.4 has the following properties: (i) All signals are bounded. (ii) The tracking error e1 converges to zero as t → ∞. (iii) If r is sufficiently rich of order 2n and Zp , Rp are coprime, then the parameter error θ˜k = θk − θ∗ converges exponentially fast to zero as k, t → ∞.

394

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

Proof First, we show that the projection algorithm used in Table 6.7 guarantees |cok | ≥ c0 ∀k ≥ 0 without affecting (i) to (iii) of the hybrid adaptive law established in Theorem 4.6.1. We rewrite the adaptive law of Table 6.7 as θk+1

=

p θk+1

=

4

4

∆p =

p θk+1 + ∆p Z tk+1 θk + Γ ²(τ )φp (τ )dτ tk ( 0 if |cp0(k+1) | ≥ c0 p γ0 γ00 (c0 − c0(k+1) )sgn(c0 ) otherwise

(6.5.48)

p p where cp0(k+1) is the last element of the vector θk+1 . We can view θk+1 , θk+1 as ∗ the pre- and post-projection estimate of θ respectively. It is obvious, from the definition of γ0 , γ00 that c0(k+1) = c0 sgn(c0 ) if |cp0(k+1) | < c0 and c0(k+1) = cp0(k+1) if |cp0(k+1) | ≥ c0 . Therefore, the constraint c0k ≥ c0 is satisfied for all k. Now consider the same Lyapunov function used in proving Theorem 4.6.1, i.e.,

V (k) = θ˜k> Γ−1 θ˜k we have p> p p> V (k + 1) = θ˜k+1 Γ−1 θ˜k+1 + 2θ˜k+1 Γ−1 ∆p + (∆p)> Γ−1 ∆p

(6.5.49)

In the proof of Theorem 4.6.1, we have shown that the first term in (6.5.49) satisfies Z p> p θ˜k+1 Γ−1 θ˜k+1 ≤ V (k) − (2 − Ts λm )

tk+1

²2 (τ )m2 (τ )dτ

(6.5.50)

tk

For simplicity, let us consider the case sgn(c0 ) = 1. Exactly the same analysis can be carried out when sgn(c0 ) = −1. Using Γ−1 Γ = I and the definition of γ0 , γ00 , we have   0  0      Γ−1 γ0 =  ...     0  1 Therefore, for cp0(k+1) < c0 , the last two terms in (6.5.49) can be expressed as p> 2θ˜k+1 Γ−1 ∆p + (∆p)> Γ−1 ∆p =

=

1 p 1 (c − c∗0 )(c0 − cp0 ) + (c − cp0(k+1) )2 γ00 0(k+1) γ00 0 1 (c − c∗0 )(c0 + cp0(k+1) − 2c∗0 ) < 0 γ00 0

6.5. DIRECT MRAC WITH NORMALIZED ADAPTIVE LAWS

395

where the last inequality follows because c∗0 > c0 > cp0(k+1) . For cp0(k+1) ≥ c0 , we p> have 2θ˜k+1 Γ−1 ∆p + (∆p)> Γ−1 ∆p = 0. Hence, p> 2θ˜k+1 Γ−1 ∆p + (∆p)> Γ−1 ∆p ≤ 0, ∀k ≥ 0

(6.5.51)

Combining (6.5.49)-(6.5.51), we have 4

Z

tk+1

∆V (k) = V (k + 1) − V (k) ≤ −(2 − Ts λm )

²2 (τ )m2 (τ )dτ

tk

which is similar to (4.6.9) established for the hybrid adaptive law without projection. Therefore, the projection does not affect the ideal asymptotic properties of the hybrid adaptive law. The rest of the stability proof is similar to that of Theorem 6.5.1 and is briefly outlined as follows: Noting that θ(t) = θk , ∀t ∈ [kTs , (k + 1)Ts ) is a piecewise constant function with discontinuities at t = kTs , k = 0, 1, . . . , and not differentiable, we make the following change in the proof to accommodate the discontinuity in ¯ + (θ(t) − θ(t)) ¯ ¯ is obtained by linearly interθk . We write θ(t) = θ(t) where θ(t) polating θk , θk+1 on the interval [kTs , (k + 1)Ts ). Because θk ∈ L∞ , ∆θk ∈ L2 , ¯˙ we can show that θ¯ has the following properties: (i) θ¯ is continuous, (ii) θ(t) = T ˙ ¯ θk+1 − θk ∀t ∈ [kTs , (k + 1)Ts ), and θ¯ ∈ L∞ L2 , (iii) |θ(t) − θ(t)| ≤ |θk+1 − θk | T ¯ and |θ(t) − θ(t)| ∈ L∞ L2 . Therefore, we can use θ¯ in the place of θ and the ¯ > ω, which has an L2 gain because error resulting from this substitution is (θ − θ) ¯ ∈ L2 . The rest of the proof is then the same as that for the continuous |θ − θ| ¯ > ω appears in the equations. This scheme except that an additional term (θ − θ) term, however, doesnot affect the stability analysis since it has an L2 gain. 2

Remark 6.5.1 In the analysis of the MRAC schemes in this section we assume that the nonlinear differential equations describing the stability properties of the schemes possess a unique solution. This assumption is essential for the validity of our analysis. We can establish that these differential equations do possess a unique solution by using the results on existence and uniqueness of solutions of adaptive systems given in [191].

6.5.4

Effect of Initial Conditions

The analysis of the direct MRAC schemes with normalized adaptive laws presented in the previous sections is based on the assumption that the ini-

396

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

tial conditions for the plant and reference model are equal to zero. This assumption allows us to use transfer function representations and other I/O tools that help improve the clarity of presentation. Nonzero initial conditions introduce exponentially decaying to zero terms in the parametric models (6.5.40) and (6.5.41) as follows: e1 = Wm (s)ρ∗ [up − θ∗> ω] + ²t z = θ∗> φp + ²t

(6.5.52)

where ²t = Cc> (sI − Ac )−1 e(0) and Ac , e(0) are as defined in Section 6.4. Because Ac is a stable matrix, the properties of the adaptive laws based on (6.5.52) with ²t = 0 remain unchanged when ²t 6= 0 as established in Section 4.3.7. Similarly, the ²t terms also appear in (6.5.42) and (6.5.43) as follows: 1 ˜> θ ω) + ²t c∗0 1 ˜> = G−1 p (s)Wm (s)(r + ∗ θ ω) + ²t c0

yp = Wm (s)(r + up

(6.5.53)

where in this case ²t denotes exponentially decaying to zero terms because of nonzero initial conditions. The exponentially decaying term only contributes to the constant in the inequality m2f ≤ c + ²t +

c ∗ kmf k2 + α02n ck˜ g mf k2 α02

where g˜ ∈ L2 . Applying the B-G Lemma to the above inequality we can establish, as in the case of ²t = 0, that mf ∈ L∞ . Using mf ∈ L∞ , the rest of the analysis follows using the same arguments as in the zero initial condition case.

6.6

Indirect MRAC

In the previous sections, we used the direct approach to develop stable MRAC schemes for controlling a wide class of plants with unknown parameters. The assumption on the plant and the special form of the controller enabled us to obtain appropriate parameterizations for the unknown controller vector θ∗ that in turn allows us to develop adaptive laws for estimating the controller parameter vector θ(t) directly.

6.6. INDIRECT MRAC

397

An alternative way of controlling the same class of plants is to use the indirect approach, where the high frequency gain kp and coefficients of the plant polynomials Zp (s), Rp (s) are estimated and the estimates are, in turn, used to determine the controller parameter vector θ(t) at each time t. The MRAC schemes based on this approach are referred to as indirect MRAC schemes since θ(t) is estimated indirectly using the plant parameter estimates. The block diagram of an indirect MRAC scheme is shown in Figure 6.15. The coefficients of the plant polynomials Zp (s), Rp (s) and high frequency gain kp are represented by the vector θp∗ . The on-line estimate θp (t) of θp∗ , generated by an adaptive law, is used to calculate the controller parameter vector θ(t) at each time t using the same mapping f : θp 7→ θ as the mapping f : θp∗ 7→ θ∗ defined by the matching equations (6.3.12), (6.3.16), (6.3.17). The adaptive law generating θp may share the same filtered values of up , yp , i.e., ω1 , ω2 as the control law leading to some further interconnections not shown in Figure 6.15. In the following sections we develop a wide class of indirect MRAC schemes that are based on the same assumptions and have the same stability properties as their counterparts direct MRAC schemes developed in Sections 6.4 and 6.5.

6.6.1

Scalar Example

Let us consider the plant x˙ = ax + bu

(6.6.1)

where a, b are unknown constants and sgn(b) is known. It is desired to choose u such that all signals in the closed-loop plant are bounded and the plant state x tracks the state xm of the reference model x˙ m = −am xm + bm r

(6.6.2)

where am > 0, bm and the reference input signal r are chosen so that xm (t) represents the desired state response of the plant. Control Law As in Section 6.2.2, if the plant parameters a, b were known, the control law u = −k ∗ x + l∗ r (6.6.3)

398

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

ym

- Wm (s)

r

− ?e 1 Σl+ yp 6

¡ µ + l up - Gp (s) Σ + + µ£± 6 ¡ ¡ ? ? ¡ £+ −1 ¡ £ (sI − F ) g (sI − F )−1 g

- c0 ¡ ¡

θ1> ¤¡

¤¡

¡ µ ¾ ω1

¡

¤¡

θ2>

¡ µ ω2 ¾ ¡ µ ¾ θ3

¡ ¡

Controller Parameter Plant ¾ u Calculation Using ¾θp Parameter ¾ p yp Mapping of Estimation (6.3.12),(6.3.16),(6.3.17) Figure 6.15 Block diagram of an indirect MRAC scheme.

with

bm am + a , l∗ = (6.6.4) b b could be used to meet the control objective. In the unknown parameter case, we propose u = −k(t)x + l(t)r (6.6.5) k∗ =

where k(t), l(t) are the on-line estimates of k ∗ , l∗ at time t, respectively. In direct adaptive control, k(t), l(t) are generated directly by an adaptive law. In indirect adaptive control, we follow a different approach. We evaluate k(t), l(t) by using the relationship (6.6.4) and the estimates a ˆ, ˆb of the unknown parameters a, b as follows: k(t) =

am + a ˆ(t) bm , l(t) = ˆb(t) ˆb(t)

(6.6.6)

6.6. INDIRECT MRAC

399

where a ˆ, ˆb are generated by an adaptive law that we design. Adaptive Law The adaptive law for generating a ˆ, ˆb is obtained by following the same procedure as in the identification examples of Chapter 4, i.e., we rewrite (6.6.1) as 1 x= [(a + am )x + bu] s + am and generate x ˆ, the estimate of x, from x ˆ=

1 [(ˆ a + am )x + ˆbu] = xm s + am

(6.6.7)

where the last equality is obtained by using (6.6.5), (6.6.6). As in Section 6.2.2, the estimation error ²1 = x − xm = e1 is the same as the tracking error and satisfies the differential equation e˙ 1 = −am e1 − a ˜x − ˜bu where

4 a ˜=a ˆ − a,

(6.6.8)

˜b 4 = ˆb − b

are the parameter errors. Equation (6.6.8) motivates the choice of Ã

˜2 ˜b2 1 2 a e1 + + V = 2 γ1 γ2

!

(6.6.9)

for some γ1 , γ2 > 0, as a potential Lyapunov-like function candidate for (6.6.8). The time derivative of V along any trajectory of (6.6.8) is given by

Hence, for

˜b˜b˙ a ˜a ˜˙ V˙ = −am e21 − a ˜xe1 − ˜bue1 + + γ1 γ2

(6.6.10)

˙ ˙ a ˜˙ = a ˆ˙ = γ1 e1 x, ˜b = ˆb = γ2 e1 u

(6.6.11)

we have V˙ = −am e21 ≤ 0 which implies that e1 , a ˆ, ˆb ∈ L∞ and that e1 ∈ L2 by following the usual arguments. Furthermore, xm , e1 ∈ L∞ imply that x ∈ L∞ . The boundedness of u, however, cannot be established unless we show that k(t), l(t)

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

are bounded. The boundedness of 1ˆ and therefore of k(t), l(t) cannot be b guaranteed by the adaptive law (6.6.11) because (6.6.11) may generate estimates ˆb(t) arbitrarily close or even equal to zero. The requirement that ˆb(t) is bounded away from zero is a controllability condition for the estimated plant that the control law (6.6.5) is designed for. One method for avoiding ˆb(t) going through zero is to modify the adaptive law for ˆb(t) so that adaptation takes place in a closed subset of R1 which doesnot include the zero element. Such a modification is achieved by using the following a priori knowledge: The sgn(b) and a lower bound b0 > 0 for |b| is known

(A2)

Applying the projection method with the constraint ˆb sgn(b) ≥ b0 to the adaptive law (6.6.11), we obtain    γ e u ˙ 2 1 ˙a ˆ ˆ = γ1 e1 x, b =  

0

if |ˆb| > b0 or if |ˆb| = b0 and e1 u sgn(b) ≥ 0 otherwise

(6.6.12)

where ˆb(0) is chosen so that ˆb(0)sgn(b) ≥ b0 . Analysis It follows from (6.6.12) that if ˆb(0)sgn(b) ≥ b0 , then whenever ˆb(t)sgn(b) = |ˆb(t)| becomes equal to b0 we have ˆb˙ ˆb ≥ 0 which implies that |ˆb(t)| ≥ b0 , ∀t ≥ 0. Furthermore the time derivative of (6.6.9) along the trajectory of (6.6.8), (6.6.12) satisfies (

V˙ =

−am e21 if |ˆb| > b0 or |ˆb| = b0 and e1 u sgn(b) ≥ 0 2 −am e1 − ˜be1 u if |ˆb| = b0 and e1 u sgn(b) < 0

Now for |ˆb| = b0 , we have (ˆb − b)sgn(b) < 0. Therefore, for |ˆb| = b0 and e1 u sgn(b) < 0, we have ˜be1 u = (ˆb − b)e1 u = (ˆb − b)sgn(b)(e1 usgn(b)) > 0 which implies that V˙ ≤ −am e21 ≤ 0,

∀t ≥ 0

Therefore, the function V given by (6.6.9) is a Lyapunov function for the system (6.6.8), (6.6.12) since u, x in (6.6.8) can be expressed in terms of e1 and xm where xm (t) is treated as an arbitrary bounded function of time.

6.6. INDIRECT MRAC

401

Reference Model - bm s + am Plant ¡ µ b r - l(t) + l Σ s−a ¡ µ − 6 ¡ ¡ ¡ ¾ k(t) ¢

− ? e1 Σl + x 6

Adaptive ¾ - Law for a ˆ, ˆb ¾ (6.6.12)

¡ ¡

k(t) l(t)

xm

ˆb a ˆ? ? Controller Parameter Calculation (6.6.6)

Figure 6.16 Block diagram for implementing the indirect MRAC scheme given by (6.6.5), (6.6.6), and (6.6.12).

Hence the equilibrium e1e = 0, a ˆe = a, ˆbe = b is u.s. and e1 , ˆb, a ˆ ∈ L∞ . Using the usual arguments, we have e1 ∈ L2 and e˙ 1 ∈ L∞ which imply that ˙ e1 (t) = x(t) − xm (t) → 0 as t → ∞ and therefore that a ˆ˙ (t), ˆb(t) → 0 as t → ∞. As in the direct case it can be shown that if the reference input signal ˜ ˜l converge to r(t) is sufficiently rich of order 2 then ˜b, a ˜ and, therefore, k, zero exponentially fast. Implementation The proposed indirect MRAC scheme for (6.6.1) described by (6.6.5), (6.6.6), and (6.6.12) is implemented as shown in Figure 6.16.

6.6.2

Indirect MRAC with Unnormalized Adaptive Laws

As in the case of direct MRAC considered in Section 6.5, we are interested in extending the indirect MRAC scheme for the scalar plant of Section 6.6.1 to a higher order plant. The basic features of the scheme of Section 6.6.1 is that the adaptive law is driven by the tracking error and a single Lyapunov function is used to design the adaptive law and establish signal boundedness.

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

In this section, we extend the results of Section 6.6.1 to plants with relative degree n∗ = 1. The same methodology is applicable to the case of n∗ ≥ 2 at the expense of additional algebraic manipulations. We assign these more complex cases as problems for the ambitious reader in the problem section. Let us start by considering the same plant and control objective as in the direct MRAC scheme of Section 6.4.1 where the relative degree of the plant is assumed to be n∗ = 1. We propose the same control law ω˙ 1 = F ω1 + gup ,

ω1 (0) = 0

ω˙ 2 = F ω2 + gyp ,

ω2 (0) = 0

(6.6.13)

>

up = θ ω as in the direct MRAC case where θ(t) is calculated using the estimate of kp and the estimates of the coefficients of the plant polynomials Zp (s), Rp (s), represented by the vector θp (t), at each time t. Our goal is to develop an adaptive law that generates the estimate θp (t) and specify the mapping from θp (t) to θ(t) that allows us to calculate θ(t) at each time t. We start with h

i>

the mapping that relates the unknown vectors θ∗ = θ1∗> , θ2∗> , θ3∗ , c∗0 and ∗ θp specified by the matching equations (6.3.12), (6.3.16), and (6.3.17) (with Q(s) = 1 due to n∗ = 1), i.e., c∗0 =

km kp

θ1∗> α(s) = Λ(s) − Zp (s), Rp (s) − Λ0 (s)Rm (s) θ2∗> α(s) + θ3∗ Λ(s) = kp

(6.6.14)

To simplify (6.6.14) further, we express Zp (s), Rp (s), Λ(s), Λ0 (s)Rm (s) as Zp (s) = sn−1 + p> 1 αn−2 (s) Rp (s) = sn + an−1 sn−1 + p> 2 αn−2 (s) Λ(s) = sn−1 + λ> αn−2 (s) Λ0 (s)Rm (s) = sn + rn−1 sn−1 + ν > αn−2 (s) > > where p1, p2 ∈Rn−1, an−1 are the plant parameters, i.e., θp∗ = [kp , p> 1 , an−1, p2 ] ; n−1 λ, ν ∈ R and rn−1 are the coefficients of the known polynomials Λ(s),

6.6. INDIRECT MRAC

403 £

¤>

Λ0 (s)Rm (s) and αn−2 (s) = sn−2 , sn−3 , . . . , s, 1 in (6.6.14) to obtain the equations

, which we then substitute

km kp = λ − p1 p2 − an−1 λ + rn−1 λ − ν = kp an−1 − rn−1 = kp

c∗0 = θ1∗ θ2∗ θ3∗

(6.6.15)

If we let kˆp (t), pˆ1 (t), pˆ2 (t), a ˆn−1 (t) be the estimate of kp , p1 , p2 , an−1 respectively at each time t, then θ(t) = [θ1> , θ2> , θ3 , c0 ]> may be calculated as km ˆ kp (t) θ1 (t) = λ − pˆ1 (t) pˆ2 (t) − a ˆn−1 (t)λ + rn−1 λ − ν θ2 (t) = kˆp (t) c0 (t) =

θ3 (t) =

(6.6.16)

a ˆn−1 (t) − rn−1 kˆp (t)

provided |kˆp (t)| = 6 0, ∀t ≥ 0. The adaptive laws for generating pˆ1 , pˆ2 , a ˆn−1 , kˆp on-line can be developed by using the techniques of Chapter 4. In this section we concentrate on adaptive laws that are driven by the tracking error e1 rather than the normalized estimation error, and are developed using the SPR-Lyapunov design approach. We start with the parametric model given by equation (6.4.6), i.e., e1 = Wm (s)ρ∗ (up − θ∗> ω) (6.6.17) k

where ρ∗ = c1∗ = kmp . As in the direct case, we choose Wm (s), the transfer 0 function of the reference model, to be SPR with relative degree n∗ = 1. > The adaptive law for θp = [kˆp , pˆ> ˆn−1 , pˆ> 1 ,a 2 ] is developed by first relating e1 with the parameter error θ˜p = θp − θp∗ through the SPR transfer function Wm (s) and then proceeding with the Lyapunov design approach as follows:

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

We rewrite (6.6.17) as e1 = Wm (s)

´ 1 ³ kp up − kp θ∗> ω − kˆp up + kˆp θ> ω km

(6.6.18)

where −kˆp up + kˆp θ> ω = 0 because of (6.6.13). If we now substitute for kp θ∗ , kˆp θ from (6.6.15) and (6.6.16), respectively, in (6.6.18) we obtain e1= Wm (s)

´ ³ ´ i 1 h˜ ³ > > > > ˆ kp λ ω1−up + p˜> ω +˜ a y −λ ω − k p ˆ ω +k p ω n−1 p 2 p 1 1 p 1 1 2 2 km

4 4 4 ˆn−1 − an−1 are the parameter ˜n−1 = a where k˜p = kˆp − kp , p˜2 = pˆ2 − p2 , a > > > ˜ ˆ> ω1 − kp p˜> ω1 ˆ errors. Because −kp pˆ1 ω1 + kp p1 ω1 + kp pˆ1 ω1 − kp pˆ> 1 1 1 ω1 = −kp p we have i 1 h˜ > e1 = Wm (s) kp ξ1 + a ˜n−1 ξ2 + p˜> ω − k p ˜ ω (6.6.19) 2 p 1 2 1 km

where 4

4

> ξ1 = λ> ω1 − up − pˆ> 1 ω1 , ξ2 = yp − λ ω2 ,

4

p˜1 = pˆ1 − p1

A minimal state-space representation of (6.6.19) is given by h

e˙ = Ac e + Bc k˜p ξ1 + a ˜n−1 ξ2 + p˜> ˜> 2 ω2 − kp p 1 ω1

i

e1 = Cc> e

(6.6.20)

where Cc> (sI − Ac )−1 Bc = Wm (s) k1m . Defining the Lyapunov-like function V =

k˜p2 a ˜2 p˜> Γ−1 p˜1 p˜> Γ−1 p˜2 e> Pc e + + n−1 + 1 1 |kp | + 2 2 2 2γp 2γ1 2 2

where Pc = Pc> > 0 satisfies the algebraic equations of the LKY Lemma, γ1 , γp > 0 and Γi = Γ> i > 0, i = 1, 2, it follows that by choosing the adaptive laws a ˆ˙ n−1 = −γ1 e1 ξ2 pˆ˙ 1 = Γ1 e1 ω1 sgn(kp ) pˆ˙ = −Γ2 e1 ω2 2

˙ kˆp =

   −γp e1 ξ1  

0

if |kˆp | > k0 or if |kˆp | = k0 and e1 ξ1 sgn(kp ) ≤ 0 otherwise

(6.6.21)

6.6. INDIRECT MRAC

405

where kˆp (0) sgn(kp ) ≥ k0 > 0 and k0 is a known lower bound for |kp |, we have V˙ =

 > qq > > Lc   −e 2 e − νc e 2 e  

>

−e> qq2 e − νc e> L2c e + e1 ξ1 k˜p

if |kˆp | > 0 or if |kˆp | = k0 and e1 ξ1 sgn(kp ) ≤ 0 if |kˆp | = k0 and e1 ξ1 sgn(kp ) > 0

where the scalar νc > 0, matrix Lc = L> c > 0 and vector q are defined in the LKY Lemma. Because for e1 ξ1 sgn(kp ) > 0 and |kˆp | = k0 we have (kˆp − kp ) sgn(kp ) < 0 and e1 ξ1 k˜p < 0, it follows that e> Lc e V˙ ≤ −νc 2 which implies that e1 , e, a ˆn−1 , pˆ1 , pˆ2 , kˆp ∈ L∞ and e, e1 ∈ L2 . As in Section 6.4.1, e1 ∈ L∞ implies that yp , ω1 , ω2 ∈ L∞ , which, together with θ ∈ L∞ (guaranteed by (6.6.16) and the boundedness of θp , ˆ1 ), implies that up ∈ kp L∞ . Therefore, all signals in the closed-loop system are bounded. The convergence of e1 to zero follows from e1 ∈ L2 and e˙ 1 ∈ L∞ guaranteed by (6.6.18). We summarize the stability properties of above scheme, whose main equations are listed in Table 6.8, by the following theorem. Theorem 6.6.1 The indirect MRAC scheme shown in Table 6.8 guarantees that all signals are u.b., and the tracking error e1 converges to zero as t → ∞. We should note that the properties of Theorem 6.6.1 are established under the assumption that the calculation of θ(t) is performed instantaneously. This assumption is quite reasonable if we consider implementation with a fast computer. However, we can relax this assumption by using a hybrid adaptive law to update θp at discrete instants of time thus providing sufficient time for calculating θ(t). The details of such a hybrid scheme are left as an exercise for the reader. As in the case of the direct MRAC schemes with unnormalized adaptive laws, the complexity of the indirect MRAC without normalization increases with the relative degree n∗ of the plant. The details of such schemes for the case of n∗ = 2 and higher are given as exercises in the problem section.

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

Table 6.8 Indirect MRAC scheme with unnormalized adaptive law for n∗ = 1 Z (s)

Plant

yp = kp Rpp (s) up , n∗ = 1

Reference model

Zm (s) ym = Wm (s)r, Wm (s) = km R m (s)

Control law

ω˙ 1 = F ω1 + gup ω˙ 2 = F ω2 + gyp up =h θ> ω i θ = θ1> , θ2> , θ3 , c0 ; ω = [ω1> , ω2> , yp , r]> if |kˆp | > k0 or ˆ if |kp | = k0 and e1 ξ1 sgn(kp ) ≤ 0   0 otherwise ˙a ˆn−1 = −γ1 e1 ξ2 pˆ˙ 1 = Γ1 e1 ω1 sgn(kp ) pˆ˙ 2 = −Γ2 e1 ω2 e1 = yp − ym > ξ1 = λ> ω1 − up − pˆ> 1 ω1 ; ξ2 = yp − λ ω2 ˙ kˆp =

Adaptive law

Calculation of θ(t)

Design variables

  −γp e1 ξ1

c0 (t) = km /kˆp (t) θ1 (t) = λ − pˆ1 (t) θ2 (t) = (ˆ p2 (t) − a ˆn−1 (t)λ + rn−1 λ − ν)/kˆp (t) θ3 (t) = (ˆ an−1 (t) − rn−1 )/kˆp (t) k0 : lower bound for |kp | ≥ k0 > 0; λ ∈ Rn−1 : coefficient vector of Λ(s) − sn−1 ; rn−1 ∈ R1 : coefficient of sn−1 in Λ0 (s)Rm (s); ν ∈ Rn−1 : coefficient vector of Λ0 (s)Rm (s) − sn − rn−1 sn−1 ; Λ(s), Λ0 (s) as defined in Section 6.4.1

6.6. INDIRECT MRAC

407

Another interesting class of indirect MRAC schemes with unnormalized adaptive laws is developed in [116, 117, 118] using a systematic recursive procedure, called backstepping. The procedure is based on a specific state space representation of the plant and leads to control and adaptive laws that are highly nonlinear.

6.6.3

Indirect MRAC with Normalized Adaptive Law

As in the direct MRAC case, the design of indirect MRAC with normalized adaptive laws is conceptually simple. The simplicity arises from the fact that the control and adaptive laws are designed independently and are combined using the certainty equivalence approach. As mentioned earlier, in indirect MRAC the adaptive law is designed to provide on-line estimates of the high frequency gain kp and of the coefficients of the plant polynomials Zp (s), Rp (s) by processing the plant input and output measurements. These estimates are used to compute the controller parameters at each time t by using the relationships defined by the matching equations (6.3.12), (6.3.16), (6.3.17). The adaptive law is developed by first expressing the plant in the form of a linear parametric model as shown in Chapters 2, 4, and 5. Starting with the plant equation (6.3.2) that we express in the form yp =

bm sm + bm−1 sm−1 + · · · + b0 up sn + an−1 sn−1 + · · · + a0

where bm = kp is the high frequency gain, and using the results of Section 2.4.1 we obtain the following plant parametric model: z = θp∗> φ where sn z= yp , Λp (s)

"

> (s) α> (s) αn−1 φ= up , − n−1 yp Λp (s) Λp (s)

(6.6.22) #>

θp∗ = [0, . . . , 0, bm , · · · , b0 , an−1 , . . . , a0 ]> | {z } n−m−1 > and Λp (s) = sn + λ> p αn−1 (s) with λp = [λn−1 , . . . , λ0 ] is a Hurwitz polynomial. Since in this case m is known, the first n − m − 1 elements of θp∗ are known to be equal to zero.

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

The parametric model (6.6.22) may be used to generate a wide class of adaptive laws by using Tables 4.2 to 4.5 from Chapter 4. Using the estimate θp (t) of θp∗ , the MRAC law may be formed as follows: The controller parameter vectors θ1 (t), θ2 (t), θ3 (t), c0 (t) in the control law α(s) α(s) up = θ1> up + θ2> yp + θ3 yp + c0 r = θ> ω (6.6.23) Λ(s) Λ(s) h

>

i>

>

(s) (s) where ω = αΛ(s) up , αΛ(s) yp , yp , r , α(s) = αn−2 (s) and θ = [θ1> , θ2> , θ3 , c0 ]> is calculated using the mapping θ(t) = f (θp (t)). The mapping f (·) is obtained by using the matching equations (6.3.12), (6.3.16), (6.3.17), i.e.,

c∗0 =

km kp

θ1∗> α(s) = Λ(s) − Zp (s)Q(s) Q(s)Rp (s) − Λ0 (s)Rm (s) θ2∗> α(s) + θ3∗ Λ(s) = kp

(6.6.24)

where Q(s) is the quotient of Λ0RRpm and Λ(s) = Λ0 (s)Zm (s). That is, if 4 ˆ¯ (s, t) are the estimated values of the polynomials R (s), Z¯ (s) = ˆ p (s, t), Z R p p p kp Zp (s) respectively at each time t, then c0 , θ1 , θ2 , θ3 are obtained as solutions to the following polynomial equations: c0 =

km kˆp

θ1> α(s) = Λ(s) − θ2> α(s) + θ3 Λ(s) =

1 ˆ¯ ˆ t) Z p (s, t) · Q(s, ˆ kp

(6.6.25)

1 ˆ ˆ p (s, t) − Λ0 (s)Rm (s)] [Q(s, t) · R kˆp

ˆ t) is the quotient of provided kˆp 6= 0, where Q(s,

Λ0 (s)Rm (s) ˆ p (s,t) . R

Here A(s, t) ·

B(s, t) denotes the frozen time product of two operators A(s, t), B(s, t). ˆ¯ (s, t) are evaluated from the estimate ˆ p (s, t), Z The polynomials R p θp = [0, . . . , 0, ˆbm , . . . , ˆb0 , a ˆn−1 , . . . , a ˆ0 ]> | {z } n−m−1

of

θp∗ ,

i.e., ˆ p (s, t) = sn + a R ˆn−1 sn−1 + · · · + a ˆ0

6.6. INDIRECT MRAC

409

m−1 ˆ¯ (s, t) = ˆb sm + ˆb Z + · · · + ˆb0 p m m−1 s

kˆp = ˆbm As in Section 6.6.2, the estimate ˆbm = kˆp should be constrained from going through zero by using projection. The equations of the indirect MRAC scheme are described by (6.6.23), (6.6.25) where θ is generated by any adaptive law from Tables 4.2 to 4.5 based on the parametric model (6.6.22). Table 6.9 summarizes the main equations of an indirect MRAC scheme with the gradient algorithm as the adaptive law. Its stability properties are summarized by the following theorem: Theorem 6.6.2 The indirect MRAC scheme summarized in Table 6.9 guarantees that all signals are u.b., and the tracking error e1 = yp − ym converges to zero as t → ∞. The proof of Theorem 6.6.2 is more complex than that for a direct MRAC scheme due to the nonlinear transformation θp 7→ θ. The details of the proof are given in Section 6.8. The number of filters required to generate the signals ω, φ in Table 6.9 may be reduced from 4n − 2 to 2n by selecting Λp (s) = (s + λ0 )Λ(s) for some λ0 > 0 and sharing common signals in the control and adaptive law. Remark 6.6.1 Instead of the gradient algorithm, a least-squares or a hybrid adaptive law may be used in Table 6.9. The hybrid adaptive law will simplify the computations considerably since the controller parameter vector θ will be calculated only at discrete points of time rather than continuously. The indirect MRAC scheme has certain advantages over the corresponding direct scheme. First, the order of the adaptive law in the indirect case is n + m + 1 compared to 2n in the direct case. Second, the indirect scheme allows us to utilize any apriori information about the plant parameters to initialize the parameter estimates or even reduce the order of the adaptive law further as indicated by the following example. Example 6.6.1 Consider the third order plant yp =

1 up + a)

s2 (s

(6.6.26)

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

Table 6.9 Indirect MRAC scheme with normalized adaptive law Plant Reference model

yp = Gp (s)up , Gp (s) = Z¯p (s)/Rp (s), Z¯p (s) = kp Zp (s) Zm (s) ym = km R r m (s)

up = θ> ω h

ω = ω1> , ω2> , yp , r

i>

h

, θ = θ1> , θ2> , θ3 , c0

i>

Control law

ω1 =

Adaptive law

¯ ˙ pˆ˙ 1 = Γ 1 φ1 ², pˆ2 = Γ2 φ2 ² ˆp | > k0 or   γm ²φ1m if |k ˙ ˆ ˆ kp = if |kp | = k0 and φ1m ²sgn(kp ) ≥ 0   0 otherwise kˆp (0)sgn(kp ) ≥ k0 > 0 ² = (z − zˆ)/m2 , z = yp + λ> p φ2 > > zˆ = θp> φ, m2 = 1 + φ> φ, φ = [φ> 1 , φ2 ] > ˆ> θp = [0, . . . , 0, kˆp , pˆ> 1 ,p 2]

αn−2 (s) Λ(s) up ,

ω2 =

αn−2 (s) Λ(s) yp

| {z } n−m−1

(s) (s) up , φ2 = − αΛn−1 yp φ1 = αΛn−1 p (s) p (s) > > > n−m ¯ ¯ φ1 = [φ0 , φ1 ] , φ0 ∈ R , φ1 ∈ Rm 1 φ1m ∈ R is the last element of φ0 pˆ1 = [ˆbm−1 , . . . , ˆb0 ]> , pˆ2 = [ˆ an−1 , . . . , a ˆ0 ]> ˆ¯ (s, t) = kˆ sm+ pˆ> α n ˆ> α ˆ Z p p 1 m−1 (s), Rp (s, t) = s + p 2 n−1 (s) km c0 (t) = ˆ kp (t) ˆ¯ (s, t) · Q(s, ˆ t) θ> (t)α (s) = Λ(s) − 1 Z

Calculation of θ

Design variables

n−2 1 ˆp (t) p k > θ2 (t)αn−2 (s) + θ3 (t)Λ(s) ˆ t) · R ˆ p (s, t) − = ˆ1 (Q(s, kp

Λ0 (s)Rm (s)) ˆ t) = quotient of Λ0 (s)Rm (s)/R ˆ p (s, t) Q(s, k0 : lower bound for |kp | ≥ k0 > 0; Λp (s): monic Hurwitz of degree n; For simplicity, Λp (s) = (s + λ0 )Λ(s), λ0 > 0; > m×m , Λ(s) = Λ0 (s)Zm (s); Γ1 = Γ> 1 > 0, Γ2 = Γ2 > 0; Γ1 ∈ R n×n n Γ2 ∈ R ; λp ∈ R is the coefficient vector of Λp (s)−sn

6.6. INDIRECT MRAC

411

where a is the only unknown parameter. The output yp is required to track the output of ym of the reference model ym =

1 r (s + 2)3

The control law is given by up = θ11

s 1 s 1 up +θ12 up +θ21 yp +θ22 yp +θ3 yp +c0 r 2 2 2 (s + λ1 ) (s + λ1 ) (s + λ1 ) (s + λ1 )2

where θ = [θ11 , θ12 , θ21 , θ22 , θ3 , c0 ]> ∈ R6 . In direct MRAC, θ is generated by a sixth-order adaptive law. In indirect MRAC, θ is calculated from the adaptive law as follows: Using Table 6.9, we have θp = [0, 0, 1, a ˆ, 0, 0]> a ˆ˙ = γa φa ² ²=

z − zˆ , zˆ = θp> φ, z = yp + λ> p φ2 1 + φ> φ

> > φ = [φ> 1 , φ 2 ] , φ1 =

[s2 , s, 1]> [s2 , s, 1]> up , φ2 = − yp 3 (s + λ1 ) (s + λ1 )3

where Λp (s) is chosen as Λp (s) = (s + λ1 )3 , λp = [3λ1 , 3λ21 , λ31 ]> . φa = [0, 0, 0, 1, 0, 0, ]φ = −

s2 yp (s + λ1 )3

and γa > 0 is a constant. The controller parameter vector is calculated as · ¸ s ˆ t) c0 = 1, θ1> = (s + λ1 )2 − Q(s, 1 · θ2>

s 1

¸ ˆ t) · [s3 + a + θ3 (s + λ1 )2 = Q(s, ˆs2 ] − (s + λ1 )2 (s + 2)3

ˆ t) is the quotient of where Q(s,

(s+λ1 )2 (s+2)3 . s3 +ˆ as 2

5

The example demonstrates that for the plant (6.6.26), the indirect scheme requires a first order adaptive law whereas the direct scheme requires a sixthorder one.

412

6.7

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

Relaxation of Assumptions in MRAC

The stability properties of the MRAC schemes of the previous sections are based on assumptions P1 to P4, given in Section 6.3.1. While these assumptions are shown to be sufficient for the MRAC schemes to meet the control objective, it has often been argued whether they are also necessary. We have already shown in the previous sections that assumption P4 can be completely relaxed at the expense of a more complex adaptive law, therefore P4 is no longer necessary for meeting the MRC objective. In this section we summarize some of the attempts to relax assumptions P1 to P3 during the 1980s and early 1990s.

6.7.1

Assumption P1: Minimum Phase

This assumption is a consequence of the control objective in the known parameter case that requires the closed-loop plant transfer function to be equal to that of the reference model. Since this objective can only be achieved by cancelling the zeros of the plant and replacing them by those of the reference model, Zp (s) has to be Hurwitz, otherwise zero-pole cancellations in C + will take place and lead to some unbounded state variables within the closed-loop plant. The assumption of minimum phase in MRAC has often been considered as one of the limitations of adaptive control in general, rather than a consequence of the MRC objective, and caused some confusion to researchers outside the adaptive control community. One of the reasons for such confusion is that the closed-loop MRAC scheme is a nonlinear dynamic system and zero-pole cancellations no longer make much sense. In this case, the minimum phase assumption manifests itself as a condition for proving that the plant input is bounded by using the boundedness of other signals in the closed-loop. For the MRC objective and the structures of the MRAC schemes presented in the previous sections, the minimum phase assumption seems to be not only sufficient, but also necessary for stability. If, however, we modify the MRC objective not to include cancellations of unstable plant zeros, then it seems reasonable to expect to be able to relax assumption P1. For example, if we can restrict ourselves to changing only the poles of the plant and tracking a restricted class of signals whose internal model is known, then we may be able to allow plants with unstable zeros. The details of such designs

6.7. RELAXATION OF ASSUMPTIONS IN MRAC

413

that fall in the category of general pole placement are given in Chapter 7. It has often been argued that if we assume that the unstable zeros of the plant are known, we can include them to be part of the zeros of the reference model and design the MRC or MRAC scheme in a way that allows only the cancellation of the stable zeros of the plant. Although such a design seems to be straightforward, the analysis of the resulting MRAC scheme requires the incorporation of an adaptive law with projection. The projection in turn requires the knowledge of a convex set in the parameter space where the estimation is to be constrained. The development of such a convex set in the higher order case is quite awkward, if possible. The details of the design and analysis of MRAC for plants with known unstable zeros for discrete-time plants are given in [88, 194]. The minimum phase assumption is one of the main drawbacks of MRAC for the simple reason that the corresponding discrete-time plant of a sampled minimum phase continuous-time plant is often nonminimum phase [14].

6.7.2

Assumption P2: Upper Bound for the Plant Order

The knowledge of an upper bound n for the plant order is used to determine the order of the MRC law. This assumption can be completely relaxed if the MRC objective is modified. For example, it has been shown in [102, 159, 160] that the control objective of regulating the output of a plant of unknown order to zero can be achieved by using simple adaptive controllers that are based on high-gain feedback, provided the plant is minimum phase and the plant relative degree n∗ is known. The principle behind some of these highgain stabilizing controllers can be explained for a minimum phase plant with n∗ = 1 and arbitrary order as follows: Consider the following minimum phase plant with relative degree n∗ = 1: yp =

kp Zp (s) up Rp (s)

From root locus arguments, it is clear that the input up = −θyp sgn(kp ) with sufficiently large gain θ will force the closed loop characteristic equation Rp (s) + |kp |θZp (s) = 0

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

to have roots in Re [s] < 0. Having this result in mind, it can be shown that the adaptive control law up = − θyp sgn (kp ) ,

θ˙ = yp2

can stabilize any minimum phase plant with n∗ = 1 and of arbitrary order. As n∗ increases, the structure and analysis of the high gain adaptive controllers becomes more complicated [159, 160]. Another class of adaptive controllers for regulation that attracted considerable interest in the research community is based on search methods and discontinuous adjustments of the controller gains [58, 137, 147, 148]. Of particular theoretical interest is the controller proposed in [137] referred to as universal controller that is based on the rather weak assumption that only the order nc of a stabilizing linear controller needs to be known for the stabilization and regulation of the output of the unknown plant to zero. The universal controller is based on an automated dense search throughout the set of all possible nc -order linear controllers until it passes through a subset of stabilizing controllers in a way that ensures asymptotic regulation and the termination of the search. One of the drawbacks of the universal controller is the possible presence of large overshoots as pointed out in [58] which limits its practicality. An interesting MRAC scheme that is also based on high gain feedback and discontinuous adjustment of controller gains is given in [148]. In this case, the MRC objective is modified to allow possible nonzero tracking errors that can be forced to be less than a prespecified (arbitrarily small) constant after an (arbitrarily short) prespecified period of time, with an (arbitrarily small) prespecified upper bound on the amount of overshoot. The only assumption made about the unknown plant is that it is minimum phase. The adaptive controllers of [58, 137, 147, 148] where the controller gains are switched from one constant value to another over intervals of times based on some cost criteria and search methods are referred to in [163] as nonidentifier-based adaptive controllers to distinguish them from the class of identifier-based ones that are studied in this book.

6.7.3

Assumption P3: Known Relative Degree n∗

The knowledge of the relative degree n∗ of the plant is used in the MRAC schemes of the previous sections in order to develop control laws that are free

6.7. RELAXATION OF ASSUMPTIONS IN MRAC

415

of differentiators. This assumption may be relaxed at the expense of additional complexity in the control and adaptive laws. For the identifier-based schemes, several approaches have been proposed that require the knowledge of an upper bound n∗u for n∗ [157, 163, 217]. In the approach of [163], n∗u parameterized controllers Ci , i = 1, 2, . . . , n∗u are constructed in a way that Ci can meet the MRC objective for a reference model Mi of relative degree i when the unknown plant has a relative degree i. A switching logic with hysteresis is then designed that switches from one controller to another based on some error criteria. It is established in [165] that switching stops in finite time and the MRC objective is met exactly. In another approach given in [217], the knowledge of an upper bound n∗u and lower bound n∗l of n∗ are used to construct a feedforward dynamic term that replaces co r in the standard MRC law, i.e., up is chosen as up = θ1>

α(s) α(s) + θ2> + θ3 yp + θ4> b1 (s)n1 (s)Wm (s)r Λ(s) Λ(s)

h

∗

i>

∗

where b1 (s) = 1, s, . . . , sn¯ ,n ¯ ∗ = n∗u − n∗l , θ4∗ ∈ Rn¯ +1 and n1 (s) is an arbitrary monic Hurwitz polynomial of degree n∗l . The relative degree of the transfer function Wm (s) of the reference model is chosen to be equal to n∗u . It can be shown that for some constant vectors θ1∗ , θ2∗ , θ3∗ , θ4∗ the MRC objective is achieved exactly, provided θ4∗ is chosen so that kp θ4∗> b1 (s) is a monic Hurwitz polynomial of degree n∗ − n∗l ≤ n ¯ ∗ , which implies that the last n ¯ ∗ − n∗ − n∗l elements of θ∗ are equal to zero. The on-line estimate θi of θi∗ , i = 1, . . . , 4 is generated by an adaptive law designed by following the procedure of the previous sections. The adaptive law for θ4 , however, is modified using projection so that θ4 is constrained to be inside a convex set C which guarantees that kp θ4> b1 (s) is a monic Hurwitz polynomial at each time t. The development of such set is trivial when the uncertainty in the relative degree, i.e., n ¯ ∗ is less or equal to 2. For uncertainties greater than 2, the calculation of C, however, is quite involved.

6.7.4

Tunability

The concept of tunability introduced by Morse in [161] is a convenient tool for analyzing both identifier and nonidentifier-based adaptive controllers and for discussing the various questions that arise in conjunction with assumptions

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

P1 to P4 in MRAC. Most of the adaptive control schemes may be represented by the equations x˙ = A(θ)x + B (θ) r ²1 = C(θ)x

(6.7.1)

where θ : R+ 7→ Rn is the estimated parameter vector and ²1 is the estimation or tuning error. Definition 6.7.1 [161, 163] The system (6.7.1) is said to be tunable on a subset S ⊂ Rn if for each θ ∈ S and each bounded input r, every possible trajectory of the system for which ²1 (t) = 0, t ∈ [0, ∞) is bounded. Lemma 6.7.1 The system (6.7.1) is tunable on S if and only if {C(θ),A(θ)} is detectable for each θ ∈ S. If {C(θ), A(θ)} is not detectable, then it follows that the state x may grow unbounded even when adaptation is successful in driving ²1 to zero by adjusting θ. One scheme that may exhibit such a behavior is a MRAC of the type considered in previous sections that is designed for a nonminimum-phase plant. In this case, it can be established that the corresponding system of equations is not detectable and therefore not tunable. The concept of tunability may be used to analyze the stability properties of MRAC schemes by following a different approach than those we discussed in the previous sections. The details of this approach are given in [161, 163] where it is used to analyze a wide class of adaptive control algorithms. The analysis is based on deriving (6.7.1) and establishing that {C(θ), A(θ)} is detectable, which implies tunability. Detectability guarantees the existence of a matrix H(θ) such that for each fixed θ ∈ S the matrix Ac (θ) = A(θ) − H(θ)C(θ) is stable. Therefore, (6.7.1) may be written as x˙ = [A (θ) − H(θ)C(θ)] x + B(θ)r + H(θ)²1

(6.7.2)

by using the so called output injection. Now from the properties of the adap˙ ²1 ∈ L2 where m = 1 + (C > x)2 for some vector tive law that guarantees θ, 0 m C0 , we can establish that the homogeneous part of (6.7.2) is u.a.s., which, together with the B-G Lemma, guarantees that x ∈ L∞ . The boundedness of x can then be used in a similar manner as in the previous sections to establish the boundedness of all signals in the closed loop and the convergence of the tracking error to zero.

6.8. STABILITY PROOFS OF MRAC SCHEMES

6.8 6.8.1

417

Stability Proofs of MRAC Schemes Normalizing Properties of Signal mf

In Section 6.5, we have defined the fictitious normalizing signal mf 4

m2f = 1 + kup k2 + kyp k2

(6.8.1)

where k · k denotes the L2δ -norm, and used its normalizing properties to establish stability of the closed-loop MRAC for the adaptive tracking and regulation examples. We now extend the results to the general SISO MRAC scheme as follows: Lemma 6.8.1 Consider the plant equation (6.3.2) and control law (6.5.39). There exists a δ > 0 such that: (i) ω1 /mf , ω2 /mf ∈ L∞ . (ii) If θ ∈ L∞ , then up /mf , yp /mf , ω/mf , W (s)ω/mf ∈ L∞ where W (s) is a proper transfer function with stable poles. (iii) If r, ˙ θ ∈ L∞ , then in addition to (i) and (ii), we have ky˙ p k/mf , kωk/m ˙ f ∈ L∞ . Proof

(i) Because ω1 =

α(s) up , Λ(s)

ω2 =

α(s) yp Λ(s)

α(s) and each element of Λ(s) has relative degree greater or equal to 1, it follows from Lemma 3.3.2 and the definition of mf that ω1 /mf , ω2 /mf ∈ L∞ . (ii) We can apply Lemma 3.3.2 to equation (6.5.42), i.e., ¶ µ 1 ˜> yp = Wm (s) r + ∗ θ ω c0

to obtain (using θ ∈ L∞ ) that |yp (t)| ≤ c + ckθ˜> ωk ≤ c + ckωk where c ≥ 0 denotes any finite constant and k · k the L2δ norm. On the other hand, we have kωk ≤ kω1 k + kω2 k + kyp k + krk ≤ ckup k + ckyp k + c ≤ cmf + c y

therefore, mpf ∈ L∞ . Because ω = [ω1> , ω2> , yp , r]> and ω1 /mf , ω2 /mf , yp /mf , r/mf ∈ L∞ , it follows that ω/mf ∈ L∞ . From up = θ> ω and ω/mf , θ ∈ L∞ we 4

conclude that up /mf ∈ L∞ . Consider φ = W (s)ω. We have |φ| ≤ ckωk for some δ > 0 such that W (s − 2δ ) is stable. Hence, mφf ∈ L∞ due to kωk/mf ∈ L∞ .

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

(iii) Note that y˙ p = sWm (s)[r + c1∗ θ˜> ω] and sWm (s) is a proper stable transfer 0 function, therefore from Lemma 3.3.2 we have ky˙ p k ≤ c + ckθ˜> ωk ≤ c + cmf i.e., ky˙ p k/mf ∈ L∞ . Similarly, because kωk ˙ ≤ kω˙ 1 k + kω˙ 2 k + ky˙ p k + krk, ˙ applying ky˙ k Lemma 3.3.2 we have kω˙ 1 k ≤ ckup k, kω˙ 2 k ≤ ckyp k which together with mpf , r˙ ∈ L∞ imply that kωk/m ˙ f ∈ L∞ and the lemma is proved. In (i) to (iii), δ > 0 is chosen 1 so that Λ(s) , Wm (s), W (s) are analytic in Re[s] ≥ δ/2. 2

6.8.2

Proof of Theorem 6.5.1: Direct MRAC

In this section we present all the details of the proof in the five steps outlined in Section 6.5.3. Step 1. Express the plant input and output in terms of the adaptation error θ˜> ω. From Figure 6.14, we can verify that the transfer function between the input r + c1∗ θ˜> ω and the plant output yp is given by 0

¶ µ 1 yp = Gc (s) r + ∗ θ˜> ω c0

where Gc (s) =

(1 −

θ1∗> (sI

−

c∗0 kp Zp − kp Zp [θ2∗> (sI − F )−1 g + θ3∗ ]

F )−1 g)Rp

α(s) Using (6.3.12), (6.3.13), and (sI −F )−1g = Λ(s) , we have, after cancellation of all the stable common zeros and poles in Gc (s), that Gc (s) = Wm (s). Therefore, the plant output may be written as µ ¶ 1 yp = Wm (s) r + ∗ θ˜> ω (6.8.2) c0

Because yp = Gp (s)up and G−1 p (s) has stable poles, we have µ ¶ 1 ˜> −1 up = Gp (s)Wm (s) r + ∗ θ ω c0

(6.8.3)

where G−1 p (s)Wm (s) is stable (due to assumption P1) and biproper (due to assumption M2). For simplicity, let us denote the L2δ -norm k(·)t k2δ for some δ > 0 by 4

k · k. Using the fictitious normalizing signal m2f = 1 + kup k2 + kyp k2 , it follows from (6.8.2), (6.8.3) and Lemma 3.3.2 that m2f ≤ c + ckθ˜> ωk2

(6.8.4)

6.8. STABILITY PROOFS OF MRAC SCHEMES

419

holds for some δ > 0 such that Wm (s−δ/2)G−1 p (s−δ/2) is a stable transfer function, where c ≥ 0 in (6.8.4) and in the remainder of this section denotes any finite constant. The normalizing properties of mf have been established in Section 6.8.1, i.e., all signals in the closed-loop adaptive system and some of their derivatives are bounded from above by mf provided θ ∈ L∞ . Step 2. Use the swapping lemmas and properties of the L2δ norm to bound kθ˜> ωk from above with terms that are guaranteed by the adaptive law to have finite L2 gains. Using the Swapping Lemma A.2 in Appendix A, we can express θ˜> ω as µ > ¶ ³ ´ ˙ > > ˜ ˜ ˜ θ ω = F1 (s, α0 ) θ ω + θ ω˙ + F (s, α0 ) θ˜> ω (6.8.5) αn

∗

0) where F (s, α0 ) = (s+α00 )n∗ , F1 (s, α0 ) = 1−F (s,α , α0 > 0 is an arbitrary cons ∗ stant and n is the relative degree of Wm (s). On the other hand, using Swapping Lemma A.1, we can write ³ ´ ˜˙ θ˜> ω = W −1 (s) θ˜> W (s)ω + Wc (s)((Wb (s)ω > )θ) (6.8.6)

where W (s) is a strictly proper transfer function with poles and zeros in C − that we will specify later. Using (6.8.6) in (6.8.5) we obtain ˙> ˜˙ θ˜> ω = F1 [θ˜ ω + θ˜> ω] ˙ + F W −1 [θ˜> W (s)ω + Wc ((Wb ω > )θ)]

(6.8.7)

where F (s)W −1 (s) can be made proper by choosing W (s) appropriately. We obtain a bound for kθ˜> ωk by considering each adaptive law separately. Adaptive Law of Table 6.5. We express the normalized estimation error as ² = Wm L(ρ∗ θ˜> φ − ρ˜ξ − ²n2s ) i.e.,

1 −1 −1 θ˜> φ = θ˜> L−1 ω = ∗ (Wm L ² + ρ˜ξ + ²n2s ) ρ

(6.8.8)

Choosing W (s) = L−1 (s) and substituting for θ˜> W (s)ω = θ˜> L−1 (s)ω from (6.8.8) into (6.8.7), we obtain · ¸ 1 ρ˜ ²n2 ˙> ˙ −1 θ˜> ω = F1 (θ˜ ω + θ˜> ω) ˙ + ∗ F Wm ² + F L ∗ ξ + ∗s + Wc (Wb ω > )θ˜ ρ ρ ρ Now from the definition of ξ and using Swapping Lemma A.1 with W (s) = L−1 (s), we have ξ = uf − θ> φ = L−1 up − θ> φ, i.e., ˙ ξ = −θ> φ + L−1 θ> ω = Wc [(Wb ω > )θ]

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

Therefore, 1 1 ˙> −1 θ˜> ω = F1 [θ˜ ω + θ˜> ω] ˙ + ∗ F Wm ² + ∗ F L[ρWc (Wb ω > )θ˙ + ²n2s ] ρ ρ

(6.8.9)

From the definition of F (s), F1 (s) and Swapping Lemma A.2, it follows that for α0 > δ ∗ c −1 kF1 (s)k∞δ ≤ , kF (s)Wm (s)k∞δ ≤ cα0n α0 Therefore, kθ˜> ωk ≤

∗ c ˜˙ > (kθ ωk + kθ˜> ωk) ˙ + cα0n (k²k + k¯ ω k + k²n2s k) α0

(6.8.10)

˙ Qb = Wb (s)ω > . Using Lemma 3.3.2, we can show that where ω ¯ = ρWc (s)[Qb (t)θ], Qb /mf ∈ L∞ and, therefore, ˜˙ f k k¯ ω k ≤ ckθm (6.8.11) Using the normalizing properties of mf established in Lemma 6.8.1, we have ˙> ˙ f k, kθ˜> ωk kθ˜ ωk ≤ ckθm ˙ ≤ ckωk ˙ ≤ cmf , k²n2s k ≤ ck²ns mf k It, therefore, follows from (6.8.10), (6.8.11) that kθ˜> ωk ≤

∗ c ˙ c ˙ f k + k²ns mf k) kθmf k + mf + cα0n (k²k + kθm α0 α0

which we express as ∗

kθ˜> ωk ≤ cα0n k˜ g mf k +

c mf α0

(6.8.12)

(6.8.13)

˙ 2 , by taking α0 > 1 so that 1 ≤ αn∗ and using k²k ≤ where g˜2 = ²2 n2s + ²2 + |θ| 0 α0 ˙ k²mf k due to mf ≥ 1. Since ²ns , ², θ ∈ L2 , it follows that g˜ ∈ L2 . Adaptive Laws of Table 6.6. The normalized estimation error is given by ²=

θ˜> φp Wm (s)up − zˆ = − , n2s = φ> p φp 1 + n2s 1 + n2s

where φp = [Wm (s)ω1> , Wm (s)ω2> , Wm (s)yp , yp ]> . Let −1 ωp = [ω1> , ω2> , yp , Wm yp ]>

then φp = Wm (s)ωp . To relate θ˜> ω with ², we write θ˜> ω = θ˜0> ω0 + c˜0 r

6.8. STABILITY PROOFS OF MRAC SCHEMES

421

where ω0 = [ω1> , ω2> , yp ]> and θ˜0 = [θ˜1> , θ˜2> , θ˜3 ]> . From yp = Wm (r + −1 we have r = Wm yp −

1 ˜> θ ω, c∗ 0

1 ˜> θ ω) c∗0

therefore,

c˜0 c˜0 −1 θ˜> ω = θ˜0> ω0 + c˜0 Wm [yp ] − ∗ θ˜> ω = θ˜> ωp − ∗ θ˜> ω c0 c0 i.e.,

c∗ θ˜> ω = 0 θ˜> ωp c0

(6.8.14)

Using φp = Wm (s)ωp and applying Swapping Lemma A.1 we have ˙ Wm (s)θ˜> ωp = θ˜> φp + Wc (Wb ωp> )θ˜ From Swapping Lemma A.2, we have ˙> θ˜> ωp = F1 (θ˜ ωp + θ˜> ω˙ p ) + F (θ˜> ωp ) ∗

−1 where F1 (s), F (s) satisfy kF1 (s)k∞δ k ≤ αc0 , kF (s)Wm (s)k∞δ ≤ cα0n for α0 > δ. Using the above two inequalities, we obtain

˙> −1 ˜> ˜˙ θ˜> ωp = F1 (θ˜ ωp + θ˜> ω˙ p ) + F Wm [θ φp + Wc (Wb ωp> )θ]

(6.8.15)

Using (6.8.14) we obtain c0 ˙ > c0 c˙0 ˙> θ˜ ωp + θ˜> ω˙ p = ∗ θ˜> ω + ∗ θ˜ ω + ∗ θ˜> ω˙ c0 c0 c0 which we use together with θ˜> φp = −² − ²n2s to express (6.8.15) as µ ¶ c˙0 ˜> c0 ˜˙ > c0 ˜> −1 ˜˙ (6.8.16) [−²−²n2s +Wc (Wb ωp> )θ] θ˜> ωp = F1 θ ω + θ ω + θ ω ˙ +F Wm c∗0 c∗0 c∗0 Due to the boundedness of

1 c0 ,

it follows from (6.8.14) that

kθ˜> ωk ≤ ckθ˜> ωp k

(6.8.17)

Using the same arguments as we used in establishing (6.8.13) for the adaptive laws of Table 6.5, we use (6.8.16), (6.8.17) and the normalizing properties of mf to obtain ∗ c kθ˜> ωk ≤ ckθ˜> ωp k ≤ cα0n k˜ g mf k + mf α0

(6.8.18)

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

where g˜ ∈ L2 . Step 3. Use the B-G Lemma to establish signal boundedness. Using (6.8.13) or (6.8.18) in m2f ≤ c + ckθ˜> ωk2 , we have ∗

m2f ≤ c + cα02n k˜ g mf k2 +

c 2 m α02 f

(6.8.19)

which for large α0 implies ∗

m2f ≤ c + cα02n k˜ g mf k2 for some other constant c ≥ 0. The boundedness of mf follows from g˜ ∈ L2 and the B-G Lemma. Because mf , θ ∈ L∞ , it follows from Lemma 6.8.1 that up , yp , ω ∈ L∞ and therefore all signals in the closed-loop plant are bounded. Step 4. Establish convergence of the tracking error to zero. Let us now consider the equation for the tracking error that relates e1 with signals that are guaranteed by the adaptive law to be in L2 . For the adaptive law of Table 6.5(A), we have e1 = ² + Wm L²n2s + Wm Lρξ ˙ ², ²ns , θ˙ ∈ L2 and ω, ns ∈ L∞ , it Because ξ = L−1 θ> ω − θ> φ = Wc (Wb ω > )θ, follows that e1 ∈ L2 . In addition we can establish, using (6.8.2), that e˙ 1 ∈ L∞ and therefore from Lemma 3.2.5, we conclude that e1 (t) → 0 as t → ∞. −1 For the adaptive laws of Table 6.5(B,C) the proof follows by using L(s) = Wm (s) and following exactly the same arguments as above. For the adaptive laws of Table 6.6 we have ²m2 = Wm θ> ω − θ> φp Applying Swapping Lemma A.1, we have Wm θ> ω = θ> Wm ω + Wc (Wb ω > )θ˙ i.e., ²m2 = θ> [Wm ω − φp ] + Wc (Wb ω > )θ˙ Now, Wm ω − φp = [0, . . . , 0, −e1 ]> , hence, c0 e1 = −²m2 + Wc (Wb ω > )θ˙ Because c10 , m, ω are bounded and ²m, θ˙ ∈ L2 , it follows that e1 ∈ L2 . As before we can use the boundedness of the signals to show that e˙ 1 ∈ L∞ which together with e1 ∈ L2 imply that e1 (t) → 0 as t → ∞.

6.8. STABILITY PROOFS OF MRAC SCHEMES

423

Step 5. Establish parameter convergence. First, we show that φ, φp are PE if r is sufficiently rich of order 2n. From the definition of φ, we can write   (sI − F )−1 gup  (sI − F )−1 gyp   φ = H(s)    yp r where H(s) = Wm (s) if the adaptive law of Table 6.5(B) is used and H(s) = L−1 (s) if that of Table 6.5(A) is used. Using up = G−1 p (s)yp and yp = Wm (s)r + e1 , we have φ = φm + φ¯ where

   (sI − F )−1 gG−1 (sI − F )−1 gG−1 p (s)Wm (s) p (s)     (sI − F )−1 gWm (s) (sI − F )−1 g r, φ¯ = H(s) e1 φm = H(s)     Wm (s) 1 0 1 

Because e1 ∈ L2 , it follows from the properties of the PE signals that φ is PE if and only if φm is PE. In the proof of Theorem 6.4.1 and 6.4.2, we have proved that   (sI − F )−1 gG−1 p (s)Wm (s)  4  (sI − F )−1 gWm (s) r φ0 =    Wm (s) 1 is PE provided r is sufficiently rich of order 2n. Because H(s) is stable and minimum phase and φ˙ 0 ∈ L∞ owing to r˙ ∈ L∞ , it follows from Lemma 4.8.3 (iv) that φm = H(s)φ0 is PE. From the definition of φp , we have     (sI −F )−1 gG−1 Wm (s)(sI −F )−1 gG−1 p (s)Wm (s) p (s)     (sI − F )−1 gWm (s) Wm (s)(sI − F )−1 g r +  e1 φp = Wm (s)     Wm (s) Wm (s) 1 1 Because φp has the same form as φ, the PE property of φp follows by using the same arguments. We establish the convergence of the parameter error and tracking error to zero as follows: First, let us consider the adaptive laws of Table 6.5. For the adaptive law based on the SPR-Lyapunov approach (Table 6.5(A)), we have ² = Wm L(ρ∗ θ˜> φ − ρ˜ξ − ²n2s ) ˙ θ˜ = −Γ²φsgn(kp /km )

(6.8.20)

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

where Cc> (sI − Ac )−1 Bc = Wm (s)L(s) is SPR. The stability properties of (6.8.20) are established by Theorem 4.5.1 in Chapter 4. According to Theorem 4.5.1 (iii), ˜ θ(t) → 0 as t → ∞ provided φ, φ˙ ∈ L∞ , φ is PE and ξ ∈ L2 . Because ξ = Wc (Wb ω > )θ˙ and ω ∈ L∞ , θ˙ ∈ L2 , we have ξ ∈ L2 . From up , yp ∈ L∞ and the expression for φ, we can establish that φ, φ˙ ∈ L∞ . Because φ is shown to be PE, ˜ to zero follows. From θ(t) ˜ → 0 as t → ∞ and ρ˜ξ, ²n2 ∈ L2 the convergence of θ(t) s and the stability of Ac , we have that e(t) → 0 as t → ∞. In fact, the convergence of e(t) to zero follows from the properties of the Lyapunov-like function used to analyze (6.8.20) in Theorem 4.5.1 without requiring φ to be PE. The proof for the adaptive law of Table 6.5(B) follows from the above arguments by replacing L−1 (s) with Wm (s) and using the results of Theorem 4.5.2 (iii). For the adaptive laws of Table 6.6, we have established in Chapter 4 that without ˜ → 0 as t → ∞ exponentially fast provided φp is PE. Since projection projection, θ(t) can only make the derivative V˙ of the Lyapunov-like function V , used to analyze the stability properties of the adaptive law, more negative the exponential convergence of θ˜ to zero can be established by following exactly the same steps as in the case of no projection.

6.8.3

Proof of Theorem 6.6.2: Indirect MRAC

We follow the same steps as in proving stability for the direct MRAC scheme: Step 1. Express yp , up in terms of θ˜> ω. Because the control law for the indirect MRAC is the same as the direct MRAC scheme, equations (6.8.2), (6.8.3) still hold, i.e., we have µ ¶ µ ¶ 1 ˜> 1 ˜> −1 yp = Wm (s) r + ∗ θ ω , up = Gp (s)Wm (s) r + ∗ θ ω c0 c0 As in the direct case, we define the fictitious normalizing signal 4

m2f = 1 + kup k2 + kyp k2 where k · k denotes the L2δ -norm for some δ > 0. Using the same arguments as in the proof of Theorem 6.5.1, we have m2f ≤ c + ckθ˜> ωk2 Step 2. We upper bound kθ˜> ωk with terms that are guaranteed by the adaptive laws to have L2 gains. From (6.6.25), we have θ1> αn−2 (s) = Λ(s) −

1 ˆ¯ ˆ t) Z p (s, t) · Q(s, kˆp

(6.8.21)

6.8. STABILITY PROOFS OF MRAC SCHEMES θ2> αn−2 (s) + θ3 Λ(s) =

1 ˆ ˆ p (s, t) − Λ0 (s)Rm (s)] [Q(s, t) · R ˆ kp

425 (6.8.22)

Consider the above polynomial equations as operator equations. We apply (6.8.21) m (s) m (s) up , and (6.8.22) to WΛ(s) yp to obtain to the signal WΛ(s) θ1> Wm (s)ω1

= Wm (s)up −

θ2> Wm (s)ω2 + θ3 Wm (s)yp

=

1 ˆ¯ ˆ t) Wm (s) up Z p (s, t) · Q(s, Λ(s) kˆp

1 ˆ ˆ p (s, t) − Λ0 (s)Rm (s)] Wm (s) yp [Q(s, t) · R ˆ Λ(s) kp

Combining these two equations, we have θ0> Wm (s)ω0

1 ˆ¯ ˆ t) Wm (s) up Z p (s, t) · Q(s, ˆ Λ(s) kp 1 ˆ ˆ p (s, t) − Λ0 (s)Rm (s)] Wm (s) yp (6.8.23) + [Q(s, t) · R ˆ Λ(s) kp

= Wm (s)up −

where θ0 = [θ1> , θ2> , θ3 ]> , ω0 = [ω1> , ω2> , yp ]> . Repeating the same algebraic manipulation, but replacing θ, θp by θ∗ , θp∗ in the polynomial equations, we have θ0∗> Wm (s)ω0

= +

Z¯p (s)Q(s) Wm (s) up kp Λ(s) Q(s)Rp (s) − Λ0 (s)Rm (s) Wm (s) yp kp Λ(s)

Wm (s)up −

(6.8.24)

where Q(s) is the quotient of Λ0 (s)Rm (s)/Rp (s) whose order is n∗ − 1 and n∗ is the relative degree of Gp (s). Subtracting (6.8.24) from (6.8.23), we have ( ) 1 W (s) 1 W (s) m m > ˆ ˆ t) ˆ t) · R ˆ p (s, t) θ˜0 Wm (s)ω0 = − Z¯ p (s, t) · Q(s, up + Q(s, yp Λ(s) Λ(s) kˆp kˆp ( ) 1 Λ0 (s)Rm (s) 1 Λ0 (s)Rm (s) Wm (s)yp − Wm (s)yp − Λ(s) kp Λ(s) kˆp ½¯ ¾ Zp (s)Q(s) Rp (s)Q(s) + Wm (s)up − Wm (s)yp kp Λ(s) kp Λ(s) 4

=

ef − e1f + e2f

4 where θ˜0 = θ0 − θ0∗ and 4

ef = −

1 ˆ¯ ˆ t) Wm (s) up + 1 Q(s, ˆ t) · R ˆ p (s, t) Wm (s) yp Z p (s, t) · Q(s, Λ(s) Λ(s) kˆp kˆp

(6.8.25)

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

e1f , e2f are defined as the terms in the second and third brackets of (6.8.25), respectively. Because c0 = kkˆm , c∗0 = kkmp , Λ(s) = Λ0 (s)Zm (s) and Z¯p (s)up = Rp (s)yp , p we have e1f = (c0 − c∗0 )yp , e2f = 0 (6.8.26) 4

−1 Using (6.8.26) in (6.8.25) and defining ωp = [ω0> , Wm (s)yp ]> we can write

θ˜> Wm (s)ωp = ef

(6.8.27)

c∗

Because θ˜> ω = c00 θ˜> ωp , proved in Section 6.8.2 (see equation (6.8.14) ), we use Swapping Lemma A.2 to write ´ c∗ ³ ˙ θ˜> ω = 0 F1 (s, α0 )(θ˜> ωp + θ˜> ω˙ p ) + F (s, α0 )θ˜> ωp c0

(6.8.28)

where F (s, α0 ) and F1 (s, α0 ) are as defined in Section 6.8.2 and satisfy kF1 (s, α0 )k∞δ ≤

c , α0

∗

−1 kF (s, α0 )Wm (s)k∞δ ≤ cα0n

for any α0 > δ > 0. Applying Swapping Lemma A.1 to Wm (s)θ˜> ωp and using (6.8.27), we obtain ´ ³ −1 (s) θ˜> Wm (s)ωp + Wc (s)(Wb (s)ωp> )θ˙ θ˜> ωp = Wm ³ ´ −1 = Wm (s) ef + Wc (s)(Wb (s)ωp> )θ˙ (6.8.29) Substituting (6.8.29) in (6.8.28), we have ³ ´´ c∗ ³ ˙ −1 θ˜> ω = 0 F1 (s, α0 )(θ˜> ωp + θ˜> ω˙ p ) + F (s, α0 )Wm (s) ef + Wc (s)(Wb (s)ωp> )θ˙ c0 Because c0 is bounded from below, i.e., |c0 | > c0 > 0, it follows from Lemma 3.3.2 that ¶ µ > c ˙ ˙ > > n∗ ˜ ˜ ˜ ˜ kθ ωk ≤ ω θk) (6.8.30) kθ ωp k + kθ ω˙ p k) + cα0 (kef k + k¯ α0 where ω ¯ = Wb (ωp )> . From θ˜ ∈ L∞ and the normalizing properties of mf , we have kθ˜> ωk ≤

∗ c ˙ c ˙ f k) kθmf k + mf + cα0n (kef k + kθm α0 α0

(6.8.31)

We now concentrate on the term kef k in (6.8.31). Let us denote 4 > ˆ¯ (s, t) = ˆb> α (s), R ˆ t) = ˆ p (s, t) = sn + a Q(s, q αn∗ −1 (s), Z ˆ> p p m p αn−1 (s)

6.8. STABILITY PROOFS OF MRAC SCHEMES

427

> where ˆbp = [kˆp , pˆ> ˆp = pˆ2 and pˆ1 , pˆ2 are defined in Table 6.9. Treating s as the 1 ] ,a differentiation operator, using Λp (s) = Λ(s)(s + λ0 ) and Swapping Lemma A.3 (i), we have

1 ˆ¯ ˆ¯ (s, t) Wm (s)(s + λ0 ) u ˆ t) Wm (s) up = 1 Q(s, ˆ t) · Z Z p (s, t) · Q(s, p p ˆ ˆ Λ(s) Λp (s) kp kp ¶ ½ µ 1 ˆ¯ (s, t) Wm (s)(s + λ0 ) u ˆ t) Z = Q(s, (6.8.32) p p Λp (s) kˆp ¶¸ ¾ · µ Wm (s)(s + λ0 ) ˙ > > up ˆbp −q Dn∗ −2 (s) αn∗ −2 (s) αm (s) Λp (s) ˆ¯ (s, t), f = ˆ t), B(s, t) = Z (by taking A(s, t) = Q(s, p

Wm (s) Λ(s) up )

and

1 ˆ ˆ¯ (s, t) Wm (s) y Q(s, t) · R p p ˆ Λ(s) kp ½ µ ¶ 1 ˆ¯ (s, t) Wm (s)(s + λ0 ) y ˆ t) R = Q(s, (6.8.33) p p Λp (s) kˆp · µ ¶¸ ¾ Wm (s)(s + λ0 ) > > −q Dn∗ −2 (s) αn∗ −2 (s) αn−1 (s) yp a ˆ˙ p Λp (s) ˆ¯ (s, t), f = ˆ t), B(s, t) = R (by taking A(s, t) = Q(s, p (6.8.33) in the expression for ef and noting that

Wm (s) Λ(s) yp ).

Using (6.8.32),

ˆ p (s, t) Wm (s)(s + λ0 ) yp − Zˆp (s, t) Wm (s)(s + λ0 ) up = −θ˜p> Wm (s)(s + λ0 )φ R Λp (s) Λp (s) we have ef =

o 0 1 n ˜> −θp Wm (s)(s + λ0 )φ + ef kˆp

where θp , φ are as defined in Table 6.9 and µ · 0 Wm (s)(s + λ0 ) 4 > > ef = q Dn∗ −2 (s) αn∗ −2 (s) αm (s) up Λp (s) " #! ¸ ˙ ˆbp Wm (s)(s + λ0 ) > −αn−1 (s) yp Λp (s) a ˆ˙ p Using the normalizing properties of mf and the fact that yp = Wm (s)(r + and θ ∈ L∞ , we have 0 kef k ≤ ckθ˙p mf k

(6.8.34)

(6.8.35) 1 ˜> θ ω) c∗ 0

Applying Swapping Lemma A.1 and using θ˜p> φ = −²m2 , we can write ˙ θ˜p> Wm (s)(s + λ0 )φ = −Wm (s)(s + λ0 )²m2 − Wc [Wb φ> ]θ˜p

(6.8.36)

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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

Again using the normalizing properties of mf , it follows from (6.8.34), (6.8.36) that kef k ≤ c(k²mmf k + kθ˙p mf k)

(6.8.37)

for some constant c. Combining (6.8.31), (6.8.37), we have kθ˜> ωk ≤

∗ c ˙ c ˙ f k + cαn∗ (k²mmf k + kθ˙p mf k) mf + kθmf k + cα0n kθm 0 α0 α0

(6.8.38)

From (6.6.25), we can establish, using θp ∈ L∞ , kˆ1 ∈ L∞ , θ˙p ∈ L2 , that θ ∈ L∞ p and θ˙ ∈ L2 . Hence, c kθ˜> ωk ≤ mf + ck˜ g mf k α0 where g˜2 =

1 ˙ 2 |θ| α20

∗ ˙ 2 + ²2 m2 + |θ˙p |2 ) and g˜ ∈ L2 . + α02n (|θ|

Step 3. Use the B-G Lemma to establish boundedness. This step is identical to Step 3 in the proof of stability for the direct MRAC scheme and is omitted. Step 4. Convergence of the tracking error to zero. From yp = Wm (s)(r + and θ˜> ω =

c∗ 0 ˜> c0 θ ωp ,

e1

1 ˜> θ ω) c∗0

we have = =

1 1 Wm (s)θ˜> ω = Wm (s) θ˜> ωp c∗0 c0 µ ¶ 1 c˙0 > > ˜ ˜ Wm (s)θ ωp − Wc (s) (Wb (s)θ ωp ) 2 c0 c0

(6.8.39)

where the last equality is obtained by applying the Swapping Lemma A.1. Substituting θ˜p> ω from (6.8.29) in (6.8.39), we have µ ¶ ´ 1 ³ c˙0 > ˙ > ˜ e1 = ef + Wc (s)(Wb (s)ωp )θ − Wc (s) (Wb (s)θ ωp ) 2 c0 c0 ˙ ˙ Note from (6.8.35) and a ˆ˙ p , ˆbp ∈ L2 , up , yp ∈ L∞ that e0f ∈ L2 . From ²m, θ˜p ∈ L2 and the boundedness of kˆ1 , m and φ, it also follows from (6.8.34), (6.8.37) that p ˙ c˙0 ∈ L2 , it follows that e1 ∈ L2 which together with e˙ 1 ∈ L∞ ef ∈ L2 . From ef , θ, imply that e1 (t) → 0 as t → ∞.

2

6.9. PROBLEMS

6.9

429

Problems

6.1 Consider the first order plant

b u s−1 where b > 0 is the only unknown parameter. Design and analyze a direct MRAC scheme that can stabilize the plant and force y to follow the output ym of the reference model 2 r ym = s+2 for any bounded and continuous reference signal r. y=

6.2 The dynamics of a throttle to speed subsystem of a vehicle may be represented by the first-order system b V = θ+d s+a where V is the vehicle speed, θ is the throttle angle and d is a constant load disturbance. The parameters b > 0, a are unknown constants whose values depend on the operating state of the vehicle that is defined by the gear state, steady-state velocity, drag, etc. We would like to design a cruise control system by choosing the throttle angle θ so that V follows a desired velocity Vm generated by the reference model Vm =

0.5 Vs s + 0.5

where Vs is the desired velocity set by the driver. (a) Assume that a, b, and d are known exactly. Design an MRC law that meets the control objective. (b) Design and analyze a direct MRAC scheme to be used in the case of a, b, and d (with b > 0) being unknown. (c) Simulate your scheme in (b) by assuming Vs = 35 and using the following values for a, b, and d: (i) a = 0.02, b = 1.3, d = 10; (ii) a = 0.02(2 + sin 0.01t), b = 1.3, d = 10 sin 0.02t. 6.3 Consider the adaptive tracking problem discussed in Section 6.2.2, i.e., the output y of the plant x˙ = ax + bu y=x is required to track the output ym of the reference model x˙ m = −am xm + bm r,

ym = xm

where a and b are unknown constants with b 6= 0 and sign(b) unknown. Design an adaptive controller to meet the control objective.

430

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL

6.4 Consider the following plant x˙ = −x + bu where b 6= 0 is unknown. Design an adaptive controller that will force x to track xm , the state of the reference model x˙ m = −xm + r for any bounded reference input r. 6.5 Consider the following SISO plant yp = kp

Zp (s) up Rp (s)

where kp , Zp (s), and Rp (s) are known. Design an MRC scheme for yp to track ym generated by the reference model ym = km

Zm (s) r Rm (s)

where Rm (s) has the same degree as Rp (s). Examine stability when Zp (s) is Hurwitz and when it is not. Comment on your results. 6.6 Consider the third order plant yp =

kp (s + b0 ) up s3 + a2 s2 + a1 s + a0

where ai , i = 0, 1, 2; b0 , kp are constants and b0 > 0. The transfer function of the reference model is given by ym =

1 r (s + 1)2

(a) Assuming that ai , b0 , and kp are known, design an MRC law that guarantees closed-loop stability and yp → ym as t → ∞ for any bounded reference input r. (b) Repeat (a) when ai , b0 , and kp are unknown and kp > 0. (c) If in (b) a2 = 0, a1 = 0, a0 = 1 are known but kp , b0 are unknown, indicate the simplification that results in the control law. 6.7 Show that the MRC law given by (6.3.29) in Remark 6.3.6 meets the MRC objective for the plant given by (6.3.1). 6.8 Show that the MRC law given by (6.3.27) or (6.3.28) in Remark 6.3.5 meets the MRC objective for the plant (6.3.1) for any given nonzero initial conditions.

6.9. PROBLEMS

431

6.9 Repeat the proof of Theorem 6.4.2 by using a minimal state space representation of the error system e1 = Wm (s) (s + p0 ) ρ∗ θ˜> φ as explained in Remark 6.4.5. 6.10 Consider the third-order plant yp =

s3

+ a2

s2

1 up + a1 s + a0

where ai , i = 0, 1, 2 are unknown constants and the reference model ym =

2 r (s + 1)(s + 2)(s + 2.5)

(a) Design a direct MRAC law with unnormalized adaptive law so that all signals in the closed-loop plant are bounded and yp (t) → ym (t) as t → ∞ for any bounded reference input r. (b) Simulate your scheme by assuming the following values for the plant parameters a0 = 1, a1 = −1.5, a2 = 0 and examine the effect of your choice of r on parameter convergence. 6.11 Design and analyze a direct MRAC with normalized adaptive law for the plant b yp = up s+a where b > 0.5, a are unknown constants. The reference model is given by ym = (a) (b) (c) (d)

3 r s+3

Design a direct MRAC scheme based on the gradient algorithm Repeat (a) for a least-squares algorithm Repeat (a) using the SPR-Lyapunov design approach with normalization Simulate your design in (c) with and without normalization. For simulations use b = 1.2, a = −1 and r a signal of your choice.

6.12 Repeat Problem 6.11(a), for a hybrid MRAC scheme. Simulate your scheme using the values of b = 1.2, a = −1 and r a signal of your choice. 6.13 Consider the plant

(s + b0 ) up (s + a)2 where b0 > 0.2 and a are unknown constants. The reference model is given by 1 r ym = s+1 yp =

432

CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL (a) Design a direct MRAC scheme with unnormalized adaptive law (b) Repeat (a) with a normalized adaptive law (c) Design an indirect MRAC scheme with an unnormalized adaptive law (d) Repeat (c) with a normalized adaptive law (e) Simulate one direct and one indirect MRAC scheme of your choice from (a) to (d) and compare their performance when b0 = 2, a = −5, a nd r = 2 + sin0.8t. Comment.

6.14 Consider the control law (6.3.27) for the plant (6.3.1) where θi∗ , i = 1, 2, 3, c∗0 are the desired controller parameters. Design and analyze a direct MRAC scheme based on this control law. 6.15 Consider the control law given by equation (6.3.29) in Remark 6.3.6 designed for the plant (6.3.1). Design and analyze a direct MRAC scheme based on this control law. 6.16 Consider the SISO plant yp =

kp Zp (s) up Rp (s)

where Zp (s), Rp (s) are monic, Zp (s) is Hurwitz and the relative degree n∗ = 1. The order n of Rp is unknown. Show that the adaptive control law up = − θyp sgn(kp ), θ˙ = yp2 guarantees signal boundedness and convergence of yp to zero for any finite n. (Hint: The plant may be represented as x˙ 1 = A11 x1 + A12 yp , x1 ∈ Rn−1 y˙ p = A21 x1 + a0 yp + kp up where A11 is stable.) 6.17 Following the same procedure used in Section 6.6.2, derive an indirect MRAC scheme using unnormalized adaptive laws for a plant with n∗ = 2. T 6.18 Let ω ∈ L∞ be PE and e ∈ S(µ) L∞ where µ ≥ 0. Let ωµ = ω + e. Show that there exists a µ∗ > 0 such that for any µ ∈ [0, µ∗ ), ωµ is PE. 6.19 Consider the MRAC problem of Section 6.4.1. It has been shown (see Remark 6.4.3) that the nonzero initial condition appears in the error equation as e1 = Wm (s)ρ∗ (up − θ∗> ω) + Cc> (sI − Ac )−1 e(0)

6.9. PROBLEMS

433

Show that the same stability results as in the case of e(0) = 0 can be established when e(0) 6= 0 by using the new Lyapunov-like function V =

e> Pc e θ˜> Γ−1 θ˜ ∗ + |ρ | + βe> 0 P 0 e0 2 2

where e0 is the zero-input response, i.e., e˙ 0 = Ac e0 ,

e0 (0) = e(0)

P0 satisfies A> c P0 + P0 Ac = −I and β > 0 is an arbitrary positive constant.

Chapter 7

Adaptive Pole Placement Control 7.1

Introduction

In Chapter 6 we considered the design of a wide class of MRAC schemes for LTI plants with stable zeros. The assumption that the plant is minimum phase, i.e., it has stable zeros, is rather restrictive in many applications. For example, the approximation of time delays often encountered in chemical and other industrial processes leads to plant models with unstable zeros. As we discussed in Chapter 6, the minimum phase assumption is a consequence of the MRC objective that requires cancellation of the plant zeros in an effort to make the closed-loop plant transfer function equal to that of the reference model. The same assumption is also used to express the desired controller parameters in the form of a linear or bilinear parametric model, and is, therefore, crucial for parameter estimation and the stability of the overall adaptive control scheme. Another class of control schemes that is popular in the known parameter case are those that change the poles of the plant and do not involve plant zero-pole cancellations. These schemes are referred to as pole placement schemes and are applicable to both minimum and nonminimum phase LTI plants. The combination of a pole placement control law with a parameter estimator or an adaptive law leads to an adaptive pole placement control (APPC) scheme that can be used to control a wide class of LTI plants with 434

7.1. INTRODUCTION

435

unknown parameters. As in the MRAC case, the APPC schemes may be divided into two classes: The indirect APPC schemes where the adaptive law generates online estimates of the coefficients of the plant transfer function that are then used to calculate the parameters of the pole placement control law by solving a certain algebraic equation; and the direct APPC where the parameters of the pole placement control law are generated directly by an adaptive law without any intermediate calculations that involve estimates of the plant parameters. The direct APPC schemes are restricted to scalar plants and to special classes of plants where the desired parameters of the pole placement controller can be expressed in the form of the linear or bilinear parametric models. Efforts to develop direct APPC schemes for a general class of LTI plants led to APPC schemes where both the controller and plant parameters are estimated on-line simultaneously [49, 112], leading to a rather complex adaptive control scheme. The indirect APPC schemes, on the other hand, are easy to design and are applicable to a wide class of LTI plants that are not required to be minimum phase or stable. The main drawback of indirect APPC is the possible loss of stabilizability of the estimated plant based on which the calculation of the controller parameters is performed. This drawback can be eliminated by modifying the indirect APPC schemes at the expense of adding more complexity. Because of its flexibility in choosing the controller design methodology (state feedback, compensator design, linear quadratic, etc.) and adaptive law (least squares, gradient, or SPR-Lyapunov type), indirect APPC is the most general class of adaptive control schemes. This class also includes indirect MRAC as a special case where some of the poles of the plant are assigned to be equal to the zeros of the plant to facilitate the required zero-pole cancellation for transfer function matching. Indirect APPC schemes have also been known as self-tuning regulators in the literature of adaptive control to distinguish them from direct MRAC schemes. The chapter is organized as follows: In Section 7.2 we use several examples to illustrate the design and analysis of APPC. These examples are used to motivate the more complicated designs in the general case treated in the rest of the chapter. In Section 7.3, we define the pole placement control (PPC) problem for a general class of SISO, LTI plants and solve it for the

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case of known plant parameters using several different control laws. The significance of Section 7.3 is that it provides several pole placement control laws to be used together with adaptive laws to form APPC schemes. The design and analysis of indirect APPC schemes for a general class of SISO, LTI plants is presented in Section 7.4. Section 7.5 is devoted to the design and analysis of hybrid APPC schemes where the estimated plant parameters are updated at distinct points in time. The problem of stabilizability of the estimated plant model at each instant of time is treated in Section 7.6. A simple example is first used to illustrate the possible loss of stabilizability and a modified indirect APPC scheme is then proposed and analyzed. The modified scheme is guaranteed to meet the control objective and is therefore not affected by the possible loss of stabilizability during parameter estimation. Section 7.7 is devoted to stability proofs of the various theorems given in previous sections.

7.2

Simple APPC Schemes

In this section we use several examples to illustrate the design and analysis of simple APPC schemes. The important features and characteristics of these schemes are used to motivate and understand the more complicated ones to be introduced in the sections to follow.

7.2.1

Scalar Example: Adaptive Regulation

Consider the scalar plant y˙ = ay + bu

(7.2.1)

where a and b are unknown constants, and the sign of b is known. The control objective is to choose u so that the closed-loop pole is placed at −am , where am > 0 is a given constant, y and u are bounded, and y(t) converges to zero as t → ∞. If a and b were known and b 6= 0 then the control law u = −ky + r k=

a + am b

(7.2.2) (7.2.3)

7.2. SIMPLE APPC SCHEMES

437

where r is a reference input, would lead to the closed-loop plant y˙ = −am y + br

(7.2.4)

i.e., the control law described by (7.2.2) and (7.2.3) changes the pole of the plant from a to −am but preserves the zero structure. This is in contrast to MRC, where the zeros of the plant are canceled and replaced with new ones. It is clear from (7.2.4) that the pole placement law (7.2.2) and (7.2.3) with r=0 meets the control objective exactly. Let us now consider the case where a and b are unknown. As in the MRAC case, we use the certainty equivalence approach to form adaptive pole placement control schemes as follows: We use the same control law as in (7.2.2) but replace the unknown controller parameter k with its on-line ˆ The estimate kˆ may be generated in two different ways: The estimate k. direct one where kˆ is generated by an adaptive law and the indirect one where kˆ is calculated from a ˆ + am (7.2.5) kˆ = ˆb provided ˆb 6= 0 where a ˆ and ˆb are the on-line estimates of a and b, respectively. We consider each design approach separately. Direct Adaptive Regulation In this case the time-varying gain kˆ in the control law ˆ + r, u = −ky r=0 (7.2.6) is updated directly by an adaptive law. The adaptive law is developed as follows: We add and subtract the desired control input, u∗ = −ky + r with m k = a+a in the plant equation, i.e., b y˙ = ay + bu∗ − bu∗ + bu = −am y − b(kˆ − k)y + br to obtain, for r = 0, the error equation ˜ y˙ = −am y − bky

(7.2.7)

4 ˜ and regulation where k˜ = kˆ − k, that relates the parameter error term bky 1 error y through the SPR transfer function s+am . As shown in Chapter 4,

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(7.2.7) is in the appropriate form for applying the Lyapunov design approach. Choosing y 2 k˜2 | b | V = + 2 2γ it follows that for

˙ kˆ = γy 2 sgn(b)

(7.2.8)

we have V˙ = −am y 2 ≤ 0 ˜ u ∈ L∞ and y ∈ L2 . From (7.2.7) and y, k˜ ∈ L∞ we which implies that y, k, have y˙ ∈ L∞ ; therefore, using Lemma 3.2.5 we obtain y(t) → 0 as t → ∞. In summary, the direct APPC scheme in (7.2.6) and (7.2.8) guarantees signal boundedness and regulation of the plant state y(t) to zero. The scheme, however, does not guarantee that the closed-loop pole of the plant is placed at −am even asymptotically with time. To achieve such a pole m placement result, we need to show that kˆ → a+a as t → ∞. For parameter b convergence, however, y is required to be PE which is in conflict with the objective of regulating y to zero. The conflict between parameter identification and regulation or control is well known in adaptive control and cannot be avoided in general. ˆ Indirect Adaptive Regulation In this approach, the gain k(t) in the control law ˆ u = −k(t)y + r, r=0 (7.2.9) is calculated by using the algebraic equation a ˆ + am kˆ = ˆb

(7.2.10)

with ˆb 6= 0 and the on-line estimates a ˆ and ˆb of the plant parameters a and b, respectively. The adaptive law for generating a ˆ and ˆb is constructed by using the techniques of Chapter 4 as follows: We construct the series-parallel model y˙ m = −am (ym − y) + a ˆy + ˆbu

(7.2.11)

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439

then subtract (7.2.11) from the plant equation (7.2.1) to obtain the error equation e˙ = −am e − a ˜y − ˜bu (7.2.12) 4 4 4 where e = y − ym , a ˜=a ˆ − a, ˜b = ˆb − b. Using the Lyapunov-like function

˜b2 e2 a ˜2 + + 2 2γ1 2γ2

V = for some γ1 , γ2 > 0 and choosing

a ˜˙ = γ1 ey,

˜b˙ = γ2 eu

(7.2.13)

we have V˙ = −am e2 ≤ 0 which implies that e, a ˆ, ˆb ∈ L∞ and e ∈ L2 . These properties, however, ˆ do not guarantee that ˆb(t) 6= 0 ∀t ≥ 0, a condition that is required for k, given by (7.2.10), to be finite. In fact, for kˆ to be uniformly bounded, we should have |ˆb(t)| ≥ b0 > 0 ∀t ≥ 0 and some constant b0 . Because such a condition cannot be guaranteed by the adaptive law, we modify (7.2.13) by assuming that |b| ≥ b0 > 0 where b0 and sgn(b) are known a priori, and use the projection techniques of Chapter 4 to obtain a ˆ˙ = γ1 ey (

ˆb˙ =

(7.2.14)

γ2 eu if |ˆb| > b0 or if |ˆb| = b0 and sgn(b)eu ≥ 0 0 otherwise

where ˆb(0) is chosen so that ˆb(0)sgn(b) ≥ b0 . The modified adaptive law guarantees that |ˆb(t)| ≥ b0 ∀t ≥ 0. Furthermore, the time derivative V˙ of V along the solution of (7.2.12) and (7.2.14) satisfies (

V˙ =

−am e2 if |ˆb| > b0 or if |ˆb| = b0 and sgn(b)eu ≥ 0 −am e2 − ˜beu if |ˆb| = b0 and sgn(b)eu < 0

Now for |ˆb| = b0 and sgn(b)eu < 0, since |b| ≥ b0 , we have ˜beu = ˆbeu − beu = (|ˆb| − |b|)eu sgn(b) = (b0 − |b|)eu sgn(b) ≥ 0 therefore, V˙ ≤ −am e2

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Hence, e, a ˜, ˜b ∈ L∞ ; e ∈ L2 and |ˆb(t)| ≥ b0 ∀t ≥ 0, which implies that ˜ k ∈ L∞ . Substituting for the control law (7.2.9) and (7.2.10) in (7.2.11), we obtain y˙ m = −am ym , which implies that ym ∈ L∞ , ym (t) → 0 as t → ∞ and, therefore, y, u ∈ L∞ . From (7.2.12), we have e˙ ∈ L∞ , which, together with e ∈ L2 , implies that e(t) = y(t) − ym (t) → 0 as t → ∞. Therefore, it follows that y(t) = e(t) + ym (t) → 0 as t → ∞. The indirect adaptive pole placement scheme given by (7.2.9), (7.2.10), and (7.2.14) has, therefore, the same stability properties as the direct one. It has also several differences. The main difference is that the gain kˆ is updated indirectly by solving an algebraic time varying equation at each time t. According to (7.2.10), the control law (7.2.9) is designed to meet the pole placement objective for the estimated plant at each time t rather than the actual plant. Therefore, for such a design to be possible, the estimated plant has to be controllable or at least stabilizable at each time t, which implies that |ˆb(t)| 6= 0 ∀t ≥ 0. In addition, for uniform signal boundedness, ˆb should satisfy |ˆb(t)| ≥ b0 > 0 where b0 is a lower bound for |b| that is known a priori. The fact that only the sign of b is needed in the direct case, whereas the knowledge of a lower bound b0 is also needed in the indirect case motivated the authors of [46] to come up with a modified indirect scheme presented in the next section where only the sign of b is needed.

7.2.2

Modified Indirect Adaptive Regulation

In the indirect case, the form of the matching equation a − bk = −am is used to calculate kˆ from the estimates of a ˆ, ˆb by selecting kˆ to satisfy a ˆ − ˆbkˆ = −am as indicated by (7.2.10). If instead of calculating k(t) we update it using an adaptive law driven by the error ˆ + am εc = a ˆ(t) − ˆb(t)k(t) then we end up with a modified scheme that does not require the knowledge of a lower bound for |b| as shown below. The error εc is motivated from the

7.2. SIMPLE APPC SCHEMES

441

ˆ fact that as εc → 0 the estimated closed-loop pole a ˆ(t) − ˆb(t)k(t) converges ˆ to −am , the desired pole. The adaptive law for k is developed by expressing ˜ a εc in terms of k, ˜, ˜b by using am = bk − a and adding and subtracting the ˆ term bk, i.e., εc = a ˆ − ˆbkˆ + bk − a + bkˆ − bkˆ = a ˜ − ˜bkˆ − bk˜ The adaptive law for kˆ is then obtained by using the gradient method to 2 ˜ i.e., minimize ε2c w.r.t. k, 2

1 ∂εc ˙ = γbεc k˜ = −γ 2 ∂ k˜ where γ > 0 is an arbitrary constant. Because b = |b|sgn(b) and γ is arbitrary, it follows that ˙ ˙ kˆ = k˜ = γ0 εc sgn(b) for some other arbitrary constant γ0 > 0. To assure stability of the overall scheme, the adaptive laws for a ˜, ˜b in (7.2.13) are modified as shown below: 2

1 ∂εc = γ1 ey − γ1 εc a ˆ˙ = a ˜˙ = γ1 ey − γ1 2 ∂˜ a 2 ˆb˙ = ˜b˙ = γ2 eu − γ2 1 ∂εc = γ2 eu + γ2 kε ˆ c 2 ∂˜b The overall modified indirect scheme is summarized as follows: ˆ u = −ky ˙ kˆ = γ0 εc sgn(b) a ˆ˙ = γ1 ey − γ1 εc ˆb˙ = γ2 eu + γ2 kε ˆ c εc = a ˆ − ˆbkˆ + am , y˙ m Stability Analysis

e = y − ym = −am (ym − y) + a ˆy + ˆbu

Let us choose the Lyapunov-like function V =

˜b2 e2 a ˜2 k˜2 |b| + + + 2 2γ1 2γ2 2γ0

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where e = y − ym satisfies the error equation e˙ = −am e − a ˜y − ˜bu (7.2.12) The time-derivative V˙ of V along the trajectories of the overall system is given by ˜ = −am e2 − ε2 ≤ 0 V˙ = −am e2 − εc (˜ a − ˜bkˆ − bk) c which implies that e, a ˜, ˜b, k˜ ∈ L∞ ; e, εc ∈ L2 . The boundedness of ym and, therefore, of y, u is established as follows: The series-parallel model equation (7.2.11) can be rewritten as ˆ = −(am − εc )ym + εc e y˙ m = −am (ym − y) + (ˆ a − ˆbk)y T

Because e, εc ∈ L∞ L2 , it follows that ym ∈ L∞ and, therefore, y, u ∈ L∞ . From ε˙c , e˙ ∈ L∞ and e, εc ∈ L2 , we have e(t), εc (t) → 0 as t → ∞, which imply that ym (t) → 0 as t → ∞. Therefore, y(t) → 0 as t → ∞. The modified indirect scheme demonstrates that we can achieve the same stability result as in the direct case by using the same a priori knowledge about the plant, namely, the knowledge of the sign of b. For the modified scheme, the controller parameter kˆ is adjusted dynamically by using the error εc between the closed-loop pole of the estimated plant, i.e., a ˆ − ˆbkˆ and the desired pole −am . The use of εc introduces an additional gradient term to the adaptive law for a ˆ, ˆb and increases the complexity of the overall scheme. In [46] it was shown that this method can be extended to higher order plants with stable zeros.

7.2.3

Scalar Example: Adaptive Tracking

Let us consider the same plant (7.2.1) as in Section 7.2.1, i.e., y˙ = ay + bu The control objective is modified to include tracking and is stated as follows: Choose the plant input u so that the closed-loop pole is at −am ; u, y ∈ L∞ and y(t) tracks the reference signal ym (t) = c, ∀t ≥ 0, where c 6= 0 is a finite constant. Let us first consider the case where a and b are known exactly. It follows from (7.2.1) that the tracking error e = y − c satisfies e˙ = ae + ac + bu

(7.2.15)

7.2. SIMPLE APPC SCHEMES

443

Because a, b, and c are known, we can choose u = −k1 e − k2 where k1 =

(7.2.16)

a + am ac , k2 = b b

(provided b 6= 0) to obtain e˙ = −am e

(7.2.17)

It is clear from (7.2.17) that e(t) = y(t) − ym → 0 exponentially fast. Let us now consider the design of an APPC scheme to meet the control objective when a and b are constant but unknown. The certainty equivalence approach suggests the use of the same control law as in (7.2.16) but with k1 and k2 replaced with their on-line estimates kˆ1 (t) and kˆ2 (t), respectively, i.e., u = −kˆ1 (t)e − kˆ2 (t) (7.2.18) As in Section 7.2.1, the updating of kˆ1 and kˆ2 may be done directly, or indirectly via calculation using the on-line estimates a ˆ and ˆb of the plant parameters. We consider each case separately. Direct Adaptive Tracking In this approach we develop an adaptive law that updates kˆ1 and kˆ2 in (7.2.18) directly without any intermediate calculation. By adding and subtracting the desired input u∗ = −k1 e − k2 in the error equation (7.2.15), we have e˙ = ae + ac + b(−k1 e − k2 ) + b(k1 e + k2 ) + bu = −am e + b(u + k1 e + k2 ) which together with (7.2.18) imply that e˙ = −am e − b(k˜1 e + k˜2 )

(7.2.19)

4 4 where k˜1 = kˆ1 −k1 , k˜2 = kˆ2 −k2 . As in Chapter 4, equation (7.2.19) motivates the Lyapunov-like function

V =

e2 k˜12 |b| k˜22 |b| + + 2 2γ1 2γ2

(7.2.20)

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whose time-derivative V˙ along the trajectory of (7.2.19) is forced to satisfy V˙ = −am e2 by choosing

˙ kˆ1 = γ1 e2 sgn(b),

(7.2.21)

˙ kˆ2 = γ2 esgn(b)

(7.2.22)

From (7.2.20) and (7.2.21) we have that e, kˆ1 , kˆ2 ∈ L∞ and e ∈ L2 , which, in turn, imply that y, u ∈ L∞ and e(t) → 0 as t → ∞ by following the usual arguments as in Section 7.2.1. The APPC scheme (7.2.18), (7.2.22) may be written as u = −kˆ1 e − γ2 sgn(b)

Z t 0

e(τ )dτ

˙ kˆ1 = γ1 e2 sgn(b)

(7.2.23)

indicating the proportional control action for stabilization and the integral action for rejecting the constant term ac in the error equation (7.2.15). We refer to (7.2.23) as the direct adaptive proportional plus integral (PI) controller. The same approach may be repeated when ym is a known bounded signal with known y˙m ∈ L∞ . The reader may verify that in this case, the adaptive control scheme u = −kˆ1 e − kˆ2 ym − kˆ3 y˙ m ˙ kˆ1 = γ1 e2 sgn(b) ˙ ˙ kˆ2 = γ2 eym sgn(b), kˆ3 = γ3 ey˙ m sgn(b)

(7.2.24)

where e = y − ym , guarantees that all signals in the closed-loop plant in (7.2.1) and (7.2.24) are bounded, and y(t) → ym (t) as t → ∞. Indirect Adaptive Tracking In this approach, we use the same control law as in the direct case, i.e., u = −kˆ1 e − kˆ2

(7.2.25)

but with kˆ1 , kˆ2 calculated using the equations a ˆ + am , kˆ1 = ˆb

a ˆc kˆ2 = ˆb

(7.2.26)

7.2. SIMPLE APPC SCHEMES

445

provided ˆb 6= 0, where a ˆ and ˆb are the on-line estimates of the plant parameters a and b, respectively. We generate a ˆ and ˆb using an adaptive law as follows: We start with the series-parallel model e˙ m = −am (em − e) + a ˆ(e + c) + ˆbu 4

based on the tracking error equation (7.2.15) and define the error e0 = e−em , which satisfies the equation e˙ 0 = −am e0 − a ˜(e + c) − ˜bu

(7.2.27)

The following adaptive laws can now be derived by using the same approach as in Section 7.2.1: a ˆ˙ = γ1 e0 (e + c) = γ1 e0 y (7.2.28) ( γ2 e0 u if |ˆb| > b0 or if |ˆb| = b0 and sgn(b)e0 u ≥ 0 ˆb˙ = 0 otherwise where ˆb(0) satisfies ˆb(0)sgn(b) ≥ b0 . The reader may verify that the time derivative of the Lyapunov-like function V =

˜b2 e20 a ˜2 + + 2 2γ1 2γ2

along the trajectories of (7.2.27), (7.2.28) satisfies V˙ ≤ −am e20 which implies e0 , a ˜, ˜b ∈ L∞ and e0 ∈ L2 . Because of (7.2.25) and (7.2.26), we have e˙ m = −am em , and, thus, em ∈ L∞ and em (t) → 0 as t → ∞. Therefore, we conclude that e, u and e˙ 0 ∈ L∞ . From e0 ∈ L2 , e˙ 0 ∈ L∞ we have e0 (t) → 0 as t → ∞, which implies that e(t) → 0 as t → ∞. As in Section 7.2.2, the assumption that a lower bound b0 for b is known can be relaxed by modifying the indirect scheme (7.2.25) to (7.2.28). In this case both kˆ1 and kˆ2 are adjusted dynamically rather than calculated from (7.2.26). Example 7.2.1 Adaptive Cruise Control Most of today’s automobiles are equipped with the so-called cruise control system. The cruise control system is

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responsible for maintaining a certain vehicle speed by automatically controlling the throttle angle. The mass of air and fuel that goes into the combustion cylinders and generates the engine torque is proportional to the throttle angle. The driver sets the desired speed Vd manually by speeding to Vd and then switching on the cruise control system. The driver can also set the speed Vd by using the “accelerator” button to accelerate from a lower speed to Vd through the use of the cruise control system. Similarly, if the driver interrupts a previously set speed by using the brake, the “resume” button may be used to allow the cruise control system to accelerate to the preset desired speed. Because of the changes in the dynamics of vehicles that are due to mechanical wear, loads, aerodynamic drag, etc., the use of adaptive control seems to be attractive for this application. The linearized model of the throttle system with throttle angle θ as the input and speed Vs as the output is of the form V˙ s = −aVs + bθ + d

(7.2.29)

where a and b are constant parameters that depend on the speed of the vehicle, aerodynamic drag and on the type of the vehicle and its condition. The variable d represents load disturbances resulting from uphill situations, drag effects, number of people in the vehicle, road condition, etc. The uncertainty in the values of a, b, and d makes adaptive control suitable for this application. Equation (7.2.29) is of the same form as equation (7.2.15); therefore, we can derive a direct adaptive PI control scheme by following the same procedure, i.e., the adaptive cruise control system is described by the following equations: θ ˙ˆ k1 ˙ kˆ2

=

−kˆ1 V¯s − kˆ2 γ1 V¯s2

=

γ2 V¯s

=

(7.2.30)

where V¯s = Vs − Vd and Vd is the desired speed set by the driver and γ1 , γ2 > 0 are the adaptive gains. In (7.2.30), we use the a priori knowledge of sgn(b) > 0 which is available from experimental data. The direct adaptive cruise control scheme given by (7.2.30) is tested on an actual vehicle [92]. The actual response for a particular test is shown in Figure 7.1. The design of an indirect adaptive cruise control scheme follows in a similar manner by modifying the approach of Section 7.2.3 and is left as an exercise for the reader. 5

7.3. PPC: KNOWN PLANT PARAMETERS

447

54 51

speed (mph)

48 45

vehicle speed

42

speed command 39 36 33 30 27 0

5

10

15

20

25

30

35

40

45

50

30

35

40

45

50

t (sec) 35

throttle angle (deg)

30 25 20 15 10 5 0 0

5

10

15

20

25 t(sec)

7.3

PPC: Known Plant Parameters

As we demonstrated in Sections 6.6 and 7.2, an indirect adaptive control scheme consists of three parts: the adaptive law that provides on-line estimates for the plant parameters; the mapping between the plant and controller parameters that is used to calculate the controller parameters from the on-line estimates of the plant parameters; thecontrol controlsystem. law. Figure 7.1 Response of the adaptive and cruise The form of the control law and the mapping between plant parameter

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estimates and controller parameters are the same as those used in the known plant parameter case. The purpose of this section is to develop several control laws that can meet the pole placement control objective when the plant parameters are known exactly. The form of these control laws as well as the mapping between the controller and plant parameters will be used in Section 7.4 to form indirect APPC schemes for plants with unknown parameters.

7.3.1

Problem Statement

Consider the SISO LTI plant yp = Gp (s)up , Gp (s) =

Zp (s) Rp (s)

(7.3.1)

where Gp (s) is proper and Rp (s) is a monic polynomial. The control objective is to choose the plant input up so that the closed-loop poles are assigned to those of a given monic Hurwitz polynomial A∗ (s). The polynomial A∗ (s), referred to as the desired closed-loop characteristic polynomial, is chosen based on the closed-loop performance requirements. To meet the control objective, we make the following assumptions about the plant: P1. Rp (s) is a monic polynomial whose degree n is known. P2. Zp (s), Rp (s) are coprime and degree(Zp ) < n. Assumptions (P1) and (P2) allow Zp , Rp to be non-Hurwitz in contrast to the MRC case where Zp is required to be Hurwitz. If, however, Zp is Hurwitz, the MRC problem is a special case of the general pole placement problem defined above with A∗ (s) restricted to have Zp as a factor. We will explain the connection between the MRC and the PPC problems in Section 7.3.2. In general, by assigning the closed-loop poles to those of A∗ (s), we can guarantee closed-loop stability and convergence of the plant output yp to zero provided there is no external input. We can also extend the PPC objective to include tracking, where yp is required to follow a certain class of reference signals ym , by using the internal model principle discussed in Chapter 3 as follows: The reference signal ym ∈ L∞ is assumed to satisfy Qm (s)ym = 0

(7.3.2)

7.3. PPC: KNOWN PLANT PARAMETERS

449

where Qm (s), the internal model of ym , is a known monic polynomial of degree q with nonrepeated roots on the jω-axis and satisfies P3. Qm (s), Zp (s) are coprime. For example, if yp is required to track the reference signal ym = 2 + sin(2t), then Qm (s) = s(s2 + 4) and, therefore, according to P3, Zp (s) should not have s or s2 + 4 as a factor. The effect of Qm (s) on the tracking error e1 = yp − ym is explained in Chapter 3 for a general feedback system and it is analyzed again in the sections to follow. In addition to assumptions P1 to P3, let us also assume that the coefficients of Zp (s), Rp (s), i.e., the plant parameters are known exactly and propose several control laws that meet the control objective. The knowledge of the plant parameters is relaxed in Section 7.4.

7.3.2

Polynomial Approach

We consider the control law Qm (s)L(s)up = −P (s)yp + M (s)ym

(7.3.3)

where P (s), L(s), M (s) are polynomials (with L(s) monic) of degree q + n − 1, n − 1, q + n − 1, respectively, to be found and Qm (s) satisfies (7.3.2) and assumption P3. Applying (7.3.3) to the plant (7.3.1), we obtain the closed-loop plant equation Zp M yp = ym (7.3.4) LQm Rp + P Zp whose characteristic equation LQm Rp + P Zp = 0

(7.3.5)

has order 2n + q − 1. The objective now is to choose P, L such that LQm Rp + P Zp = A∗

(7.3.6)

is satisfied for a given monic Hurwitz polynomial A∗ (s) of degree 2n + q − 1. Because assumptions P2 and P3 guarantee that Qm Rp , Zp are coprime, it

450

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

follows from Theorem 2.3.1 that L, P satisfying (7.3.6) exist and are unique. The solution for the coefficients of L(s), P (s) of equation (7.3.6) may be obtained by solving the algebraic equation Sl βl = αl∗

(7.3.7)

where Sl is the Sylvester matrix of Qm Rp , Zp of dimension 2(n+q)×2(n+q) βl = [lq> , p> ]> , a∗l = [0, . . . , 0, 1, α∗> ]> | {z } q

lq = [0, . . . , 0, 1, l> ]> ∈ Rn+q | {z } q

l = [ln−2 , ln−3 , . . . , l1 , l0 ]> ∈ Rn−1 p = [pn+q−1 , pn+q−2 , . . . , p1 , p0 ]> ∈ Rn+q α∗ = [a∗2n+q−2 , a∗2n+q−3 , . . . , a∗1 , a∗0 ]> ∈ R2n+q−1 li , pi , a∗i are the coefficients of L(s) = sn−1 + ln−2 sn−2 + · · · + l1 s + l0 = sn−1 + l> αn−2 (s) P (s) = pn+q−1 sn+q−1 + pn+q−2 sn+q−2 + · · · + p1 s + p0 = p> αn+q−1 (s) A∗ (s) = s2n+q−1 +a∗2n+q−2 s2n+q−2 +· · ·+a∗1 s+a∗0 = s2n+q−1 +α∗> α2n+q−2 (s) The coprimeness of Qm Rp , Zp guarantees that Sl is nonsingular; therefore, the coefficients of L(s), P (s) may be computed from the equation βl = Sl−1 αl∗ Using (7.3.6), the closed-loop plant is described by yp =

Zp M ym A∗

(7.3.8)

Similarly, from the plant equation in (7.3.1) and the control law in (7.3.3) and (7.3.6), we obtain Rp M up = ym (7.3.9) A∗

7.3. PPC: KNOWN PLANT PARAMETERS

ym + −e P (s) - Σl 1Q m (s)L(s) − 6

up

- Gp (s)

451

yp

-

Figure 7.2 Block diagram of pole placement control. Z M

R M

Because ym ∈ L∞ and Ap ∗ , Ap ∗ are proper with stable poles, it follows that yp , up ∈ L∞ for any polynomial M (s) of degree n + q − 1. Therefore, the pole placement objective is achieved by the control law (7.3.3) without having to put any additional restrictions on M (s), Qm (s). When ym = 0, (7.3.8), (7.3.9) imply that yp , up converge to zero exponentially fast. When ym 6= 0, the tracking error e1 = yp − ym is given by e1 =

Zp M − A∗ Zp LRp ym = ∗ (M − P )ym − ∗ Qm ym ∗ A A A

(7.3.10)

For zero tracking error, (7.3.10) suggests the choice of M (s) = P (s) to null the first term in (7.3.10). The second term in (7.3.10) is nulled by using Qm ym = 0. Therefore, for M (s) = P (s), we have e1 = Z

Zp LRp [0] − ∗ [0] A∗ A

LR

Because Ap∗ , A∗p are proper with stable poles, it follows that e1 converges exponentially to zero. Therefore, the pole placement and tracking objective are achieved by using the control law Qm Lup = −P (yp − ym )

(7.3.11)

which is implemented as shown in Figure 7.2 using n + q − 1 integrators (s) to realize C(s) = QmP(s)L(s) . Because L(s) is not necessarily Hurwitz, the realization of (7.3.11) with n+q −1 integrators may have a transfer function, namely C(s), with poles outside C − . An alternative realization of (7.3.11) is obtained by rewriting (7.3.11) as up =

Λ − LQm P up − (yp − ym ) Λ Λ

(7.3.12)

452

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

ym

up

−e1 P (s) + + Σl - Σl − 6

Λ(s)

+ 6

- Gp (s)

yp

-

Λ(s) − Qm (s)L(s) ¾ Λ(s)

Figure 7.3 An alternative realization of the pole placement control. where Λ is any monic Hurwitz polynomial of degree n+q−1. The control law (7.3.12) is implemented as shown in Figure 7.3 using 2(n + q − 1) integrators m P to realize the proper stable transfer functions Λ−LQ , Λ . We summarize the Λ main equations of the control law in Table 7.1. Remark 7.3.1 The MRC law of Section 6.3.2 shown in Figure 6.1 is a special case of the general PPC law (7.3.3), (7.3.6). We can obtain the MRC law of Section 6.3.2 by choosing Qm = 1,

A∗ = Zp Λ0 Rm ,

L(s) = Λ(s) − θ1∗> αn−2 (s),

M (s) =

Rm Λ0 kp

P (s) = −(θ2∗> αn−2 (s) + θ3∗ Λ(s))

Zm r Rm where Zp , Λ0 , Rm are Hurwitz and Λ0 , Rm , kp , θ1∗ , θ2∗ , θ3∗ , r are as defined in Section 6.3.2. Λ = Λ0 Zm ,

ym = km

Example 7.3.1 Consider the plant yp =

b up s+a

where a and b are known constants. The control objective is to choose up such that the poles of the closed-loop plant are placed at the roots of A∗ (s) = (s + 1)2 and yp tracks the constant reference signal ym = 1. Clearly the internal model of ym is Qm (s) = s, i.e., q = 1. Because n = 1, the polynomials L, P, Λ are of the form L(s) = 1,

P (s) = p1 s + p0 ,

Λ = s + λ0

7.3. PPC: KNOWN PLANT PARAMETERS

453

Table 7.1 PPC law: polynomial approach Zp (s) Rp (s) up

Plant

yp =

Reference input

Qm (s)ym = 0

Calculation

Solve for L(s) = sn−1 + l>αn−2(s) and P (s) = p>αn+q−1(s) the polynomial equation L(s)Qm (s)Rp (s) + P (s)Zp (s) = A∗ (s) or solve for βl the algebraic equation Sl βl = αl∗ , where Sl is the Sylvester matrix of Rp Qm , Zp βl = [lq> , p> ]> ∈ R2(n+q) lq = [0, . . . , 0, 1, l> ]> ∈ Rn+q | {z } q

A∗ (s) = s2n+q−1 + α∗> α2n+q−2 (s) αl∗ = [0, . . . , 0, 1, α∗> ]> ∈ R2(n+q) | {z } q

Control law

Design variables

m up = Λ−LQ up − Λ e1 = yp − ym

P Λ e1

A∗ (s) is monic Hurwitz; Qm (s) is a monic polynomial of degree q with nonrepeated roots on jω axis; Λ(s) is a monic Hurwitz polynomial of degree n+q−1

where λ0 > 0 is arbitrary and p0 , p1 are calculated by solving s(s + a) + (p1 s + p0 )b = (s + 1)2 Equating the coefficients of the powers of s in (7.3.13), we obtain p1 =

2−a , b

p0 =

1 b

(7.3.13)

454

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

Equation (7.3.13) may be also written in the form of the algebraic equation (7.3.7), i.e., the Sylvester matrix of s(s + a), b is given by   1 0 0 0  a 1 0 0   Sl =   0 a b 0  0 0 0 b and



 0  1   βl =   p1  , p0

 0  1   αl∗ =   2  1 

Therefore, the PPC law is given by up

= =

· ¸ (s + λ0 − s) 2−a 1 1 up − s+ e1 s + λ0 b b s + λ0 λ0 (2 − a)s + 1 up − e1 s + λ0 b(s + λ0 )

where e1 = yp − ym = yp − 1. A state-space realization of the control law is given by φ˙ 1 φ˙ 2 up

= −λ0 φ1 + up = −λ0 φ2 + e1 1 − 2λ0 + aλ0 2−a = λ0 φ 1 − φ2 − e1 b b

5

7.3.3

State-Variable Approach

An alternative approach for deriving a PPC law is to use a state observer and state-variable feedback. We start by considering the expression e1 =

Zp (s) up − ym Rp (s)

(7.3.14)

(s) for the tracking error. Filtering each side of (7.3.14) with QQm1 (s) , where Q1 (s) is an arbitrary monic Hurwitz polynomial of degree q, and using Qm (s)ym = 0 we obtain Zp Q1 u ¯p (7.3.15) e1 = Rp Qm

7.3. PPC: KNOWN PLANT PARAMETERS where u ¯p =

Qm up Q1

455

(7.3.16)

With (7.3.15), we have converted the tracking problem into the regulation problem of choosing u ¯p to regulate e1 to zero. Let (A, B, C) be a state space realization of (7.3.15) in the observer canonical form, i.e., e˙ = Ae + B u ¯p e1 = C > e

(7.3.17)

where ¯  ¯ I ¯ n+q−1   ¯ A = −θ1∗ ¯ − − −  , B = θ2∗ , C = [1, 0, . . . , 0]> ¯ ¯ 0 

(7.3.18)

θ1∗, θ2∗ ∈Rn+q are the coefficient vectors of the polynomials Rp (s)Qm (s) − sn+q and Zp (s)Q1 (s), respectively. Because Rp Qm , Zp are coprime, any possible zero-pole cancellation in (7.3.15) between Q1 (s) and Rp (s)Qm (s) will occur in C − due to Q1 (s) being Hurwitz, which implies that (A, B) is always stabilizable. We consider the feedback law u ¯p = −Kc eˆ,

up =

Q1 u ¯p Qm

(7.3.19)

where eˆ is the state of the full-order Luenberger observer eˆ˙ = Aˆ e + Bu ¯p − Ko (C > eˆ − e1 )

(7.3.20)

and Kc and Ko are calculated from det(sI − A + BKc ) = A∗c (s)

(7.3.21)

det(sI − A + Ko C > ) = A∗o (s)

(7.3.22)

where A∗c and A∗o are given monic Hurwitz polynomials of degree n + q. The roots of A∗c (s) = 0 represent the desired pole locations of the transfer function of the closed-loop plant whereas the roots of A∗o (s) are equal to the poles of

456

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

the observer dynamics. As in every observer design, the roots of A∗o (s) = 0 are chosen to be faster than those of A∗c (s) = 0 in order to reduce the effect of the observer dynamics on the transient response of the tracking error e1 . The existence of Kc in (7.3.21) follows from the controllability of (A, B). If (A, B) is stabilizable but not controllable because of the common factors between Q1 (s), Rp (s), the solution for Kc in (7.3.21) still exists provided A∗c (s) is chosen to contain the common factors of Q1 , Rp . Because A∗c (s) is chosen based on the desired closed-loop performance requirements and Q1 (s) is an arbitrary monic Hurwitz polynomial of degree q, we can choose Q1 (s) to be a factor of A∗c (s) and, therefore, guarantee the existence of Kc in (7.3.21) even when (A, B) is not controllable. The existence of Ko in (7.3.22) follows from the observability of (C, A). Because of the special form of (7.3.17) and (7.3.18), the solution of (7.3.22) is given by Ko = α0∗ − θ1∗ where α0∗ is the coefficient vector of A∗o (s). Theorem 7.3.1 The PPC law (7.3.19) to (7.3.22) guarantees that all signals in the closed-loop plant are bounded and e1 converges to zero exponentially fast. 4

Proof We define the observation error eo = e − eˆ. Subtracting (7.3.20) from (7.3.17) we have e˙ o = (A − Ko C > )eo (7.3.23) Using (7.3.19) in (7.3.20) we obtain eˆ˙ = (A − BKc )ˆ e + Ko C > e o

(7.3.24)

Because A − Ko C > , A − BKc are stable, the equilibrium eoe = 0, eˆe = 0 of (7.3.23), (7.3.24) is e.s. in the large. Therefore eˆ, eo ∈ L∞ and eˆ(t), eo (t) → 0 as t → ∞. From eo = e − eˆ and u ¯p = −Kc eˆ, it follows that e, u ¯p , e1 ∈ L∞ and e(t), u ¯p (t), e1 (t) → 0 as t → ∞. The boundedness of yp follows from that of e1 and ym . We prove that up ∈ L∞ as follows: Because of Assumption P3 and the stability of Q1 (s), the polynomials Zp Q1 , Rp Qm have no common unstable zeros. Therefore there exists polynomials X, Y of degree n + q − 1 with X monic that satisfy the Diophantine equation Rp Qm X + Zp Q1 Y = A∗ (7.3.25) where A∗ is a monic Hurwitz polynomial of degree 2(n + q) − 1 that contains the common zeros of Q1 (s), Rp (s)Qm (s). Dividing each side of (7.3.25) by A∗ and using it to filter up , we obtain the equation Rp Qm X Q1 Y Zp up + up = up A∗ A∗

7.3. PPC: KNOWN PLANT PARAMETERS

457

Because Qm up = Q1 u ¯p and Zp up = Rp yp , we have up =

Rp XQ1 Q 1 Y Rp u ¯p + yp ∗ A A∗

Because the transfer functions operating on u ¯p , yp are proper with poles in C − , then u ¯p , yp ∈ L∞ imply that up ∈ L∞ and the proof is complete. 2

The main equations of the state variable feedback law are summarized in Table 7.2. Example 7.3.2 Let us consider the same plant and control objective as in Example 7.3.1, i.e., b Plant yp = up s+a Control Objective Choose up such that the closed-loop poles are placed at the roots of A∗ (s) = (s + 1)2 and yp tracks ym = 1. Clearly, the internal model of ym is Qm (s) = s and the tracking error e1 = yp − ym satisfies b e1 = up − ym s+a s Filtering each side of the above equation with s+1 , i.e., using Q1 (s) = s + 1, we obtain b(s + 1) e1 = u ¯p (7.3.26) (s + a)s where u ¯p =

s s+1 up .

The state-space realization of (7.3.26) is given by · ¸ · ¸ −a 1 1 e˙ = e+ b¯ up 0 0 1 e1 = [1 0]e

The control law is then chosen as follows: · ¸ · ¸ −a 1 1 Observer eˆ˙ = eˆ + b¯ up − K0 ([1 0]ˆ e − e1 ) 0 0 1 Z t (s + 1) Control Law u ¯p = −Kc eˆ, up = u ¯p = u ¯p + u ¯p (τ )dτ s 0 > where Ko = [ko1 , ko2 ] , Kc = [kc1 , kc2 ] are calculated using (7.3.21) and (7.3.22). We select the closed-loop polynomial A∗c (s) = (s + 1)2 and the observer polynomial A∗o (s) = (s + 5)2 and solve det(sI − A + BKc ) = (s + 1)2 , det(sI − A + Ko C > ) = (s + 5)2

458

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL Table 7.2 State-space pole placement control law

Plant

yp =

Reference input

Zp (s) Rp (s) up

Qm (s)ym = 0 eˆ˙ = Aˆ e +B u ¯¯ p −Ko (C > eˆ−e1 )  ¯ I ¯ n+q−1   ∗¯ A = −θ1 ¯ − − −−  , B = θ2∗ ¯ ¯ 0 C > = [1, 0, . . . , 0], e1 = yp − ym where θ1∗ , θ2∗ ∈ Rn+q are the coefficient vectors of Rp (s)Qm (s) − sn+q and Zp (s)Q1 (s), respectively

Observer

Calculation

Ko = α0∗ − θ1∗ where α0∗ is the coefficient vector of A∗o (s) − sn+q ; Kc is solved from det(sI − A + BKc ) = A∗c (s)

Control law

u ¯p = −Kc eˆ, up =

Design variables

A∗o (s), A∗c (s) are monic Hurwitz polynomials of degree n + q; Q1 (s) is a monic Hurwitz polynomial of degree q; A∗c (s) contains Q1 (s) as a factor; Qm(s) is a monic polynomial of degree q with nonrepeated roots on the jω axis

where

· A=

−a 1 0 0

¸

· , B=

1 1

Q1 (s) ¯p Qm (s) u

¸ b, C > = [1, 0]

for Ko , Kc to obtain Kc =

1 [1 − a, 1], b

Ko = [10 − a, 25]>

7.3. PPC: KNOWN PLANT PARAMETERS

459

Note that the solution for Kc holds for any a and b 6= 0. For a = 1 and b 6= 0 the pair ½· ¸ · ¸ ¾ −a 1 1 , b 0 0 1 is uncontrollable due to a zero-pole cancellation in (7.3.26) at s = −1. Because, however, A∗c (s) = (s+1)2 contains s+1 as a factor the solution of Kc still exists. The reader can verify that for A∗c (s) = (s + 2)2 (i.e., A∗c (s) doesnot have Q1 (s) = s + 1 as a factor) and a = 1, no finite value of Kc exists to satisfy (7.3.21). 5

7.3.4

Linear Quadratic Control

Another method for solving the PPC problem is using an optimization technique to design a control input that guarantees boundedness and regulation of the plant output or tracking error to zero by minimizing a certain cost function that reflects the performance of the closed-loop system. As we have shown in Section 7.3.3, the system under consideration is e˙ = Ae + B u ¯p e1 = C > e

(7.3.27)

where u ¯p is to be chosen so that the closed-loop system has eigenvalues that are the same as the zeros of a given Hurwitz polynomial A∗c (s). If the state e is available for measurement, then the control input u ¯p = −Kc e where Kc is chosen so that A − BKc is a stable matrix, leads to the closedloop system e˙ = (A − BKc )e whose equilibrium ee = 0 is exponentially stable. The existence of such Kc is guaranteed provided (A, B) is controllable (or stabilizable). In Section 7.3.3, Kc is chosen so that det(sI − A + BKc ) = A∗c (s) is a Hurwitz polynomial. This choice of Kc leads to a bounded input u ¯p that forces e, e1 to converge to zero exponentially fast with a rate that depends on the location of the eigenvalues of A − BKc , i.e., the zeros of A∗c (s). The rate of decay of e1 to

460

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

zero depends on how negative the real parts of the eigenvalues of A − BKc are. It can be shown [95] that the more negative these values are, the larger the value of Kc and, therefore, the higher the required signal energy in u ¯p . In the limit, as the eigenvalues of A − BKc are forced to −∞, the input u ¯p becomes a string of impulsive functions that restore e(t) instantaneously to zero. These facts indicate that there exists a trade-off between the rate of decay of e1 , e to zero and the energy of the input u ¯p . This trade-off motivates the following linear quadratic control problem where the control input u¯p is chosen to minimize the quadratic cost J=

Z ∞ 0

(e21 + λ¯ u2p )dt

where λ > 0, a weighting coefficient to be designed, penalizes the level of the control input signal. The optimum control input u ¯p that minimizes J is [95] u ¯p = −Kc e, Kc = λ−1 B > P

(7.3.28)

where P = P > > 0 satisfies the algebraic equation A> P + P A − P Bλ−1 B > P + CC > = 0

(7.3.29)

known as the Riccati equation. Because (A, B) is stabilizable, owing to Assumption P3 and the fact that Q1 (s) is Hurwitz, the existence and uniqueness of P = P > > 0 satisfying (7.3.29) is guaranteed [95]. It is clear that as λ → 0, a situation known as low cost of control, ||Kc || → ∞, which implies that u ¯p may become unbounded. On the other hand if λ → ∞, a situation known as high cost of control, u ¯p → 0 if the open-loop system is stable. If the open loop is unstable, then R u ¯p is the one that minimizes 0∞ u ¯2p dt among all stabilizing control laws. In this case, the real part of the eigenvalues of A − BKc may not be negative enough indicating that the tracking or regulation error may not go to zero fast enough. With λ > 0 and finite, however, (7.3.28), (7.3.29) guarantee that A − BKc is a stable matrix, e, e1 converge to zero exponentially fast, and u ¯p is bounded. The location of the eigenvalues of A − BKc depends on the particular choice of λ. For a given λ > 0, the polynomial 4

f (s) = Rp (s)Qm (s)Rp (−s)Qm (−s) + λ−1 Zp (s)Q1 (s)Zp (−s)Q1 (−s)

7.3. PPC: KNOWN PLANT PARAMETERS

461

is an even function of s and f (s) = 0 has a total of 2(n + q) roots with n + q of them in C − and the other n + q in C + . It can be shown that the poles corresponding to the closed-loop LQ control are the same as the roots of f (s) = 0 that are located in C − [6]. On the other hand, however, given a desired polynomial A∗c (s), there may not exist a λ so that det(sI − A + Bλ−1 B > P ) = A∗c (s). Hence, the LQ control solution provides us with a procedure for designing a stabilizing control input for the system (7.3.27). It doesnot guarantee, however, that the closed-loop system has the same eigenvalues as the roots of the desired polynomial A∗c (s). The significance of the LQ solution also relies on the fact that the resulting closed-loop has good sensitivity properties. As in Section 7.3.3, the state e of (7.3.28) may not be available for measurement. Therefore, instead of (7.3.28), we use u ¯p = −Kc eˆ, Kc = λ−1 B > P

(7.3.30)

where eˆ is the state of the observer equation eˆ˙ = Aˆ e + Bu ¯p − Ko (C > eˆ − e1 )

(7.3.31)

and Ko = α0∗ − θ1∗ as in (7.3.20). As in Section 7.3.3, the control input is given by Q1 (s) u ¯p (7.3.32) up = Qm (s) Theorem 7.3.2 The LQ control law (7.3.30) to (7.3.32) guarantees that all signals in the closed-loop plant are bounded and e1 (t) → 0 as t → ∞ exponentially fast. Proof As in Section 7.3.3, we consider the system described by the error equations e˙ o = (A − Ko C > )eo eˆ˙ = (A − BKc )ˆ e + Ko C > e o

(7.3.33)

where Kc = λ−1 B > P and Ko is chosen to assign the eigenvalues of A − Ko C > to the zeros of a given Hurwitz polynomial A∗o (s). Therefore, the equilibrium eoe = 0, eˆe = 0 is e.s. in the large if and only if the matrix A − BKc is stable. We establish the stability of A − BKc by considering the system e¯˙ = (A − BKc )¯ e = (A − λ−1 BB > P )¯ e

(7.3.34)

462

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

and proving that the equilibrium e¯e = 0 is e.s. in the large. We choose the Lyapunov function V (¯ e) = e¯> P e¯ where P = P > > 0 satisfies the Riccati equation (7.3.29). Then V˙ along the trajectories of (7.3.34) is given by 1 V˙ = −¯ e> CC > e¯ − λ−1 e¯> P BB > P e¯ = −(C > e¯)2 − (B > P e¯)2 ≤ 0 λ which implies that e¯e = 0 is stable, e¯ ∈ L∞ and C > e¯, B > P e¯ ∈ L2 . We now rewrite (7.3.34) as 1 e¯˙ = (A − Ko C > )¯ e + Ko C > e¯ − BB > P e¯ λ by using output injection, i.e., adding andTsubtracting the term Ko C > e¯. Because A − Ko CT> is stable and C > e¯, B > P e¯ ∈ L∞ L2 , it follows from Corollary 3.3.1 that e¯ ∈ L∞ L2 and e¯ → 0 as t → ∞. Using the results of Section 3.4.5 it follows that the equilibrium e¯e = 0 is e.s. in the large which implies that A − BKc = A − λ−1 BB > P is a stable matrix. The rest of the proof is the same as that of Theorem 7.3.1 and is omitted. 2

The main equations of the LQ control law are summarized in Table 7.3. Example 7.3.3 Let us consider the scalar plant x˙ = yp =

−ax + bup x

where a and b are known constants and b 6= 0. The control objective is to choose up to stabilize the plant and regulate yp , x to zero. In this case Qm (s) = Q1 (s) = 1 and no observer is needed because the state x is available for measurement. The control law that minimizes Z ∞ J= (yp2 + λu2p )dt 0

is given by 1 up = − bpyp λ

7.3. PPC: KNOWN PLANT PARAMETERS

463

Table 7.3 LQ control law Zp (s) Rp (s) up

Plant

yp =

Reference input

Qm (s)ym = 0

Observer

As in Table 7.2

Calculation

Solve for P = P > > 0 the equation A> P + P A − P Bλ−1 B > P + CC > = 0 A, B, C as defined in Table 7.2

Q1 (s) ¯p Qm (s) u

Control law

u ¯p = −λ−1 B > P eˆ, up =

Design variables

λ > 0 penalizes the control effort; Q1 (s), Qm (s) as in Table 7.2

where p > 0 is a solution of the scalar Riccati equation −2ap −

p2 b2 +1=0 λ

The two possible solutions of the above quadratic equation are √ √ −λa + λ2 a2 + b2 λ −λa − λ2 a2 + b2 λ p1 = , p2 = b2 b2 It is clear that p1 > 0 and p2 < 0; therefore, the solution we are looking for is p = p1 > 0. Hence, the control input is given by q   2 + b2 a −a λ  yp up = −  + b b

464

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

that leads to the closed-loop plant r x˙ = − a2 +

b2 x λ

which implies that for any finite λ > 0, we have x ∈ L∞ and x(t) → 0 as t → ∞ exponentially fast. It is clear that for λ → 0 the closed-loop eigenvalue goes to −∞. For λ → ∞ the closed-loop eigenvalue goes to −|a|, which implies that if the open-loop system is stable then the eigenvalue remains unchanged and if the open-loop system is unstable, the eigenvalue of the closed-loop system is flipped to the left half plane reflecting the minimum effort that is required to stabilize the unstable system. The reader may verify that for the control law chosen above, the cost function J becomes q 2 a2 + bλ − a J =λ x2 (0) b2 It is clear that if a > 0, i.e., the plant is open-loop stable, the cost J is less than when a < 0, i.e., the plant is open-loop unstable. More details about the LQ problem may be found in several books [16, 95, 122]. 5 Example 7.3.4 Let us consider the same plant as in Examples 7.3.1, i.e., Plant

yp =

b up s+a

Control Objective Choose up so that the closed-loop poles are stable and yp tracks the reference signal ym = 1. Tracking Error Equations The problem is converted to a regulation problem by considering the tracking error equation e1 =

b(s + 1) s u ¯p , u ¯p = up (s + a)s s+1

where e1 = yp − ym generated as shown in Example 7.3.2. The state-space representation of the tracking error equation is given by · ¸ · ¸ −a 1 1 e˙ = e+ b¯ up 0 0 1 e1 = [1, 0]e Observer The observer equation is the same as in Example 7.3.2, i.e., · ¸ · ¸ ˙eˆ = −a 1 eˆ + 1 b¯ up − Ko ([1 0]ˆ e − e1 ) 0 0 1

7.3. PPC: KNOWN PLANT PARAMETERS

465

where Ko = [10 − a, 25]> is chosen so that the observer poles are equal to the roots of A∗o (s) = (s + 5)2 . Control law The control law, according to (7.3.30) to (7.3.32), is given by u ¯p = −λ−1 [b, b]P eˆ,

up =

s+1 u ¯p s

where P satisfies the Riccati equation · ¸> · ¸ · ¸ · −a 1 −a 1 b 1 P +P −P λ−1 [b b]P + 0 0 0 0 b 0

0 0

¸ =0

(7.3.35)

where λ > 0 is a design parameter to be chosen. For λ = 0.1, a = −0.5, b = 2, the positive definite solution of (7.3.35) is · ¸ 0.1585 0.0117 P = 0.0117 0.0125 which leads to the control law u ¯p = −[3.4037 0.4829]ˆ e, up =

s+1 u ¯p s

This control law shifts the open-loop eigenvalues from λ1 = 0.5, λ2 = 0 to λ1 = −1.01, λ2 = −6.263. For λ = 1, a = −0.5, b = 2 we have ¸ · 0.5097 0.1046 P = 0.1046 0.1241 leading to u ¯p = −[1.2287 0.4574]ˆ e and the closed-loop eigenvalues λ1 = −1.69, λ2 = −1.19. Let us consider the case where a = 1, b = 1. For these values of a, b the pair ½· ¸ · ¸¾ −1 1 1 , 0 0 1 is uncontrollable but stabilizable and the open-loop plant has eigenvalues at λ1 = 0, λ2 = −1. The part of the system that corresponds to λ2 = −1 is uncontrollable. In this case, λ = 0.1 gives · ¸ 0.2114 0.0289 P = 0.0289 0.0471 and u ¯p = −[2.402 0.760]ˆ e, which leads to a closed-loop plant with eigenvalues at −1.0 and −3.162. As expected, the uncontrollable dynamics that correspond to λ2 = −1 remained unchanged. 5

466

7.4

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

Indirect APPC Schemes

Let us consider the plant given by (7.3.1), i.e., yp = Gp (s)up ,

Gp (s) =

Zp (s) Rp (s)

where Rp (s), Zp (s) satisfy Assumptions P1 and P2. The control objective is to choose up so that the closed-loop poles are assigned to the roots of the characteristic equation A∗ (s) = 0, where A∗ (s) is a given monic Hurwitz polynomial, and yp is forced to follow the reference signal ym ∈ L∞ whose internal model Qm (s), i.e., Qm (s)ym = 0 is known and satisfies assumption P3. In Section 7.3, we assume that the plant parameters (i.e., the coefficients of Zp (s), Rp (s)) are known exactly and propose several control laws that meet the control objective. In this section, we assume that Zp (s), Rp (s) satisfy Assumptions P1 to P3 but their coefficients are unknown constants; and use the certainty equivalence approach to design several indirect APPC schemes to meet the control objective. As mentioned earlier, with this approach we combine the PPC laws developed in Section 7.3 for the known parameter case with adaptive laws that generate on-line estimates for the unknown plant parameters. The adaptive laws are developed by first expressing (7.3.1) in the form of the parametric models considered in Chapter 4, where the coefficients of Zp , Rp appear in a linear form, and then using Tables 4.1 to 4.5 to pick up the adaptive law of our choice. We illustrate the design of adaptive laws for the plant (7.3.1) in the following section.

7.4.1

Parametric Model and Adaptive Laws

We consider the plant equation Rp (s)yp = Zp (s)up where Rp (s) = sn +an−1 sn−1 +· · ·+a1 s+a0 , Zp (s) = bn−1 sn−1 +· · ·+b1 s+b0 , which may be expressed in the form [sn + θa∗> αn−1 (s)]yp = θb∗> αn−1 (s)up

(7.4.1)

7.4. INDIRECT APPC SCHEMES

467

where αn−1(s) = [sn−1, . . . , s, 1]> and θa∗ = [an−1 , . . . , a0 ]> , θb∗ = [bn−1 , . . . , b0 ]> are the unknown parameter vectors. Filtering each side of (7.4.1) with Λp1(s) , where Λp (s) = sn + λn−1 sn−1 + · · · λ0 is a Hurwitz polynomial, we obtain z = θp∗> φ

(7.4.2)

where "

> (s) αn−1 α> (s) sn z= yp , θp∗ = [θb∗> , θa∗> ]> , φ = up , − n−1 yp Λp (s) Λp (s) Λp (s)

#>

Equation (7.4.2) is in the form of the linear parametric model studied in Chapter 4, thus leading to a wide class of adaptive laws that can be picked up from Tables 4.1 to 4.5 for estimating θp∗ . Instead of (7.4.1), we can also write yp = (Λp − Rp )

1 1 yp + Zp up Λp Λp

that leads to the linear parametric model yp = θλ∗> φ h

i>

(7.4.3)

where θλ∗ = θb∗> , (θa∗ − λp )> and λp = [λn−1 , λn−2 , . . . , λ0 ]> is the coefficient vector of Λp (s) − sn . Equation (7.4.3) can also be used to generate a wide class of adaptive laws using the results of Chapter 4. The plant parameterizations in (7.4.2) and (7.4.3) assume that the plant is strictly proper with known order n but unknown relative degree n∗ ≥ 1. The number of the plant zeros, i.e., the degree of Zp (s), however, is unknown. In order to allow for the uncertainty in the number of zeros, we parameterize Zp (s) to have degree n − 1 where the coefficients of si for i = m + 1, m + 2, . . . , n − 1 are equal to zero and m is the degree of Zp (s). If m < n − 1 is known, then the dimension of the unknown vector θp∗ is reduced to n + m + 1. The adaptive laws for estimating on-line the vector θp∗ or θλ∗ in (7.4.2), (7.4.3) have already been developed in Chapter 4 and are presented in Tables 4.1 to 4.5. In the following sections, we use (7.4.2) or (7.4.3) to pick up adaptive laws from Tables 4.1 to 4.5 of and combine them with the PPC laws of Section 7.3 to form APPC schemes.

468

7.4.2

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

APPC Scheme: The Polynomial Approach

Let us first illustrate the design and analysis of an APPC scheme based on the PPC scheme of Section 7.3.2 using a first order plant model. Then we consider the general case that is applicable to an nth-order plant. Example 7.4.1 Consider the same plant as in example 7.3.1, i.e., yp =

b up s+a

(7.4.4)

where a and b are unknown constants and up is to be chosen so that the poles of the closed-loop plant are placed at the roots of A∗ (s) = (s + 1)2 = 0 and yp tracks the constant reference signal ym = 1 ∀t ≥ 0. Let us start by designing each block of the APPC scheme, i.e., the adaptive law for estimating the plant parameters a and b; the mapping from the estimates of a, b to the controller parameters; and the control law. Adaptive Law We start with the following parametric model for (7.4.4) z = θp∗> φ where z=

s yp , s+λ

φ=

1 s+λ

·

up −yp

¸

· ,

θp∗ =

b a

¸ (7.4.5)

and λ > 0 is an arbitrary design constant. Using Tables 4.1 to 4.5 of Chapter 4, we can generate a number of adaptive laws for estimating θp∗ . For this example, let us choose the gradient algorithm of Table 4.2 θ˙p = Γεφ ε=

z − θp> φ , m2

(7.4.6)

m2 = 1 + φ> φ

where Γ = Γ> > 0, θp = [ˆb, a ˆ]> and a ˆ(t), ˆb(t) is the estimate of a and b respectively. Calculation of Controller Parameters As shown in Section 7.3.2, the control law Λ − LQm P up = up − e1 (7.4.7) Λ Λ can be used to achieve the control objective, where Λ(s) = s + λ0 , L(s) = 1, 4

Qm (s) = s , P (s) = p1 s + p0 , e1 = yp − ym and the coefficients p1 , p0 of P (s) satisfy the Diophantine equation s(s + a) + (p1 s + p0 )b = (s + 1)2

(7.4.8)

7.4. INDIRECT APPC SCHEMES or equivalently the algebraic  1  a   0 0

equation

 0 0 0 0  1 1 0 0   a b 0   p1 0 0 b p0

469





 0   1  =    2  1

(7.4.9)

whose solution is

2−a 1 , p0 = b b Because a and b are unknown, the certainty equivalence approach suggests the use of the same control law but with the controller polynomial P (s) = p1 s + p0 calculated by using the estimates a ˆ(t) and ˆb(t) of a and b at each time t as if they were the true parameters, i.e., P (s, t) = pˆ1 (t)s + pˆ0 (t) is generated by solving the polynomial equation s · (s + a ˆ) + (ˆ p1 s + pˆ0 ) · ˆb = (s + 1)2 (7.4.10) ˆ for pˆ1 and pˆ0 by treating a ˆ(t) and b(t) as frozen parameters at each time t, or by solving the algebraic time varying equation      1 0 0 0 0 0  a  1   1  ˆ (t) 1 0 0   =  (7.4.11)  0 a ˆ(t) ˆb(t) 0   pˆ1 (t)   2  pˆ0 (t) 1 0 0 0 ˆb(t) p1 =

for pˆ0 and pˆ1 . The solution of (7.4.10), where a ˆ(t) and ˆb(t) are treated as constants at each time t, is referred to as pointwise to distinguish it from solutions that may be obtained with s treated as a differential operator, and a ˆ(t) and ˆb(t) treated as differentiable functions of time. The Diophantine equation (7.4.10) or algebraic equation (7.4.11) has a unique solution provided that (s + a ˆ), ˆb are coprime, i.e., provided ˆb 6= 0. The solution is given by 2−a ˆ 1 pˆ1 (t) = , pˆ0 (t) = ˆb ˆb ˆ In fact for a ˆ, b ∈ L∞ to imply that pˆ0 , pˆ1 ∈ L∞ , (s+ˆ a), ˆb have to be strongly coprime, i.e., |ˆb| ≥ b0 for some constant b0 > 0. For this simple example, the adaptive law (7.4.6) can be modified to guarantee that |ˆb(t)| ≥ b0 > 0 ∀t ≥ 0 provided sgn(b) and a lower bound b0 of |b| are known as shown in Section 7.2 and previous chapters. For clarity of presentation, let us assume that the adaptive law (7.4.6) is modified using projection (as in Section 7.2) to guarantee |ˆb(t)| ≥ b0 ∀t ≥ 0 and proceed with the rest of the design and analysis. Control Law The estimates pˆ0 (t), pˆ1 (t) are used in place of the unknown p0 , p1 to form the control law µ ¶ s 1 λ0 up − pˆ1 (t) + pˆ0 (t) (yp − ym ) (7.4.12) up = s + λ0 s + λ0 s + λ0

470

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL up

-

b s+a

yp

-

- θ˙ = Γ²φ ¾ p

-

ˆb a ˆ? ? 2−a ˆ 1 pˆ1 = , pˆ0 = ˆb ˆb pˆ1 pˆ0

λ s+λ up

?

+? ¾ Σn −

?

(ˆ p1 s+ pˆ0 ) ¡ ¡ ª

1 ¾ e1 s+λ

+? ¾ym Σn −

¡ ¡ ª

Figure 7.4 Block diagram of APPC for a first-order plant. where λ0 > 0 is an arbitrary design constant. For simplicity of implementation, λ0 may be taken to be equal to the design parameter λ used in the adaptive law (7.4.6), so that the same signals can be shared by the adaptive and control law. Implementation The block diagram of the APPC scheme with λ = λ0 for the first order plant (7.4.4) is shown in Figure 7.4. For λ = λ0 , the APPC scheme may be realized by the following equations: Filters φ˙ 1 φ˙ 2

=

−λφ1 + up ,

φ1 (0) = 0

=

−λφ2 − yp ,

φ2 (0) = 0

φ˙ m z

= =

−λφm + ym , φm (0) = 0 λφ2 + yp = −φ˙ 2

Adaptive Law ˆb˙ = a ˆ˙ = ² =

½

γ1 ²φ1 0

if |ˆb| > b0 or if |ˆb| = b0 and ²φ1 sgnˆb ≥ 0 otherwise

γ2 ²φ2 z − ˆbφ1 − a ˆ φ2 m2

,

m2 = 1 + φ21 + φ22

7.4. INDIRECT APPC SCHEMES

471

Control Law up pˆ1

= λφ1 − (ˆ p1 (t)λ − pˆ0 (t))(φ2 + φm ) − pˆ1 (t)(yp − ym ) 2−a ˆ 1 = , pˆ0 = ˆb ˆb

where γ1 , γ2 , λ > 0 are design constants and ˆb(0)sgn(b) ≥ b0 . Analysis The stability analysis of the indirect APPC scheme is carried out in the following four steps: Step 1. Manipulate the estimation error and control law equations to express yp , up in terms of the estimation error ². We start with the expression for the normalized estimation error ²m2 = z − ˆbφ1 − a ˆφ2 = −φ˙ 2 − ˆbφ1 − a ˆ φ2 which implies that

φ˙ 2 = −ˆbφ1 − a ˆφ2 − ²m2

(7.4.13)

From the control law, we have up = λφ1 + pˆ1 (φ˙ 2 + φ˙ m ) + pˆ0 φ2 + pˆ0 φm Because up − λφ1 = φ˙ 1 , it follows that φ˙ 1 = pˆ1 φ˙ 2 + pˆ0 φ2 + y¯m 4 where y¯m = pˆ1 φ˙ m + pˆ0 φm . Substituting for φ˙ 2 from (7.4.13) we obtain

φ˙ 1 = −ˆ p1ˆbφ1 − (ˆ p1 a ˆ − pˆ0 )φ2 − pˆ1 ²m2 + y¯m

(7.4.14)

Equations (7.4.13), (7.4.14) form the following state space representation for the APPC scheme: ·

up −yp

x˙ = ¸ =

A(t)x + b1 (t)²m2 + b2 y¯m x˙ + λx = (A(t) + λI)x + b1 (t)²m2 + b2 y¯m

(7.4.15)

where · x=

φ1 φ2

¸

· , A(t) =

−ˆ p1ˆb pˆ0 − pˆ1 a ˆ −ˆb −ˆ a

m2 = 1 + x> x and y¯m ∈ L∞ .

·

¸ ,

b1 (t) =

−ˆ p1 −1

¸

· ,

b2 =

1 0

¸

472

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

Step 2. Show that the homogeneous part of (7.4.15) is e.s. For each fixed t, det(sI −A(t)) = (s+ˆ a)s+ˆb(ˆ p1 s+pˆ0 ) = (s+1)2 , i.e., λ(A(t)) = −1, ∀t ≥ 0. As shown T ˙ in Chapter 4, the adaptive law guarantees that ², a ˆ, ˆb ∈ L∞ ; ², ²m, a ˆ˙ , ˆb ∈ L∞ L2 . a From pˆ1 = 2−ˆ , pˆ0 = 1ˆb and ˆb−1 ∈ L∞ (because of projection), it follows that ˆ b T T ˙ pˆ1 , pˆ0 ∈ L∞ and pˆ˙ 1 , pˆ˙ 0 ∈ L∞ L2 . Hence, kA(t)k ∈ L∞ L2 which together with λ(A(t)) = −1 ∀t ≥ 0 and Theorem 3.4.11 imply that the state transition matrix Φ(t, τ ) associated with A(t) satisfies kΦ(t, τ )k ≤ k1 e−k2 (t−τ ) ,

∀t ≥ τ ≥ 0

for some constants k1 , k2 > 0. Step 3. Use the properties of the L2δ norm and B-G Lemma to establish boundedness. For simplicity, let us now denote k(·)t k2δ for some δ > 0 with k · k. Applying Lemma 3.3.3 to (7.4.15) we obtain kxk ≤ ck²m2 k + c, |x(t)| ≤ ck²m2 k + c

(7.4.16)

for any δ ∈ [0, δ1 ) where δ1 > 0 is any constant less than 2k2 , and some finite constants c ≥ 0. As in the MRAC case, we define the fictitious normalizing signal 4

m2f = 1 + kup k2 + kyp k2 From (7.4.15) we have kup k + kyp k ≤ ckxk + ck²m2 k + c, which, together with (7.4.16), implies that m2f ≤ ck²m2 k2 + c p Because |φ1 | ≤ ckup k, |φ2 | ≤ ckyp k for δ ∈ [0, 2λ), it follows that m = 1 + φ> φ ≤ cmf and, therefore, m2f ≤ ck˜ g mf k2 + c 4

where g˜ = ²m ∈ L2 because of the properties of the adaptive law, or Z m2f ≤ c

0

t

e−δ(t−τ ) g˜2 (τ )m2f (τ )dτ + c

where 0 < δ ≤ δ ∗ and δ ∗ = min[2λ, δ1 ], δ1 ∈ (0, 2k2 ). Applying the B-G Lemma, we can establish that mf ∈ L∞ . Because m ≤ cmf , we have m and therefore φ1 , φ2 , x, x, ˙ up , yp ∈ L ∞ . Step 4. Establish tracking error convergence. We consider the estimation error equation ²m2 = −φ˙ 2 − a ˆφ2 − ˆbφ1

7.4. INDIRECT APPC SCHEMES or ²m2 = (s + a ˆ)

473

1 1 yp − ˆb up s+λ s+λ 4 d dt ,

Operating on each side of (7.4.17) with s = s(²m2 ) = s(s + a ˆ)

(7.4.17)

we obtain

s 1 ˙ 1 yp − ˆb up − ˆb up s+λ s+λ s+λ

(7.4.18)

by using the property s(xy) = xy ˙ + xy. ˙ For λ = λ0 , it follows from the control law (7.4.12) that 1 s up = −(ˆ p1 s + pˆ0 ) e1 s+λ s+λ which we substitute in (7.4.18) to obtain s(²m2 ) = s(s + a ˆ)

1 1 ˙ 1 yp + ˆb(ˆ p1 s + pˆ0 ) e1 − ˆb up s+λ s+λ s+λ

(7.4.19)

1 s 1 Now, because s(s + a ˆ) s+λ yp = (s + a ˆ) s+λ yp + a ˆ˙ s+λ yp and se1 = syp − sym = syp (note that sym = 0), we have

s(s + a ˆ)

1 s 1 yp = (s + a ˆ) e1 + a ˆ˙ yp s+λ s+λ s+λ

which we substitute in (7.4.19) to obtain h i s(²m2 ) = (s + a ˆ)s + ˆb(ˆ p1 s + pˆ0 )

1 1 ˙ 1 e1 + a ˆ˙ yp − ˆb up s+λ s+λ s+λ

Using (7.4.10) we have (s + a ˆ)s + ˆb(ˆ p1 s + pˆ0 ) = (s + 1)2 and therefore s(²m2 ) = or

(s + 1)2 1 ˙ 1 e1 + a ˆ˙ yp − ˆb up s+λ s+λ s+λ

s(s + λ) 2 s+λ ˙ 1 s + λ ˆ˙ 1 ²m − a ˆ yp + b up (7.4.20) 2 2 (s + 1) (s + 1) s + λ (s + 1)2 s + λ T ˙ Because up , ypT , m, ² ∈ L∞ and a ˆ˙ , ˆb, ²m ∈ L∞ L2 , it follows from Corollary 3.3.1 that e1 ∈ L∞ L2 . Hence, if we show that e˙ 1 ∈ L∞ , then by Lemma 3.2.5 we can conclude that e1 → 0 as t → ∞. Since e˙ 1 = y˙ p = ayp + bup ∈ L∞ , it follows that e1 → 0 as t → ∞. ˙ We can continue our analysis and establish that ², a ˆ˙ , ˆb, pˆ˙ 0 , pˆ˙ 1 → 0 as t → ∞. There is no guarantee, however, that pˆ0 , pˆ1 , a ˆ, ˆb will converge to the actual values p0 , p1 , a, b respectively unless the reference signal ym is sufficiently rich of order 2, which is not the case for the example under consideration. e1 =

474

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

As we indicated earlier the calculation of the controller parameters pˆ0 (t), pˆ1 (t) at each time is possible provided the estimated plant polynomials (s + a ˆ(t)), ˆb(t) are strongly coprime, i.e., provided |ˆb(t)| ≥ b0 > 0 ∀t ≥ 0. This condition implies that at each time t, the estimated plant is strongly controllable. This is not surprising because the control law is calculated at each time t to meet the control objective for the estimated plant. As we will show in Section 7.6, the adaptive law without projection cannot guarantee that |ˆb(t)| ≥ b0 > 0 ∀t ≥ 0. Projection requires the knowledge of b0 and sgn(b) and constrains ˆb(t) to be in the region |ˆb(t)| ≥ b0 where controllability is always satisfied. In the higher-order case, the problem of controllability or stabilizability of the estimated plant is more difficult as we demonstrate below for a general nth-order plant. 5

General Case Let us now consider the nth-order plant yp =

Zp (s) up Rp (s)

where Zp (s), Rp (s) satisfy assumptions P1, P2, and P3 with the same control objective as in Section 7.3.1, except that in this case the coefficients of Zp , Rp are unknown. The APPC scheme that meets the control objective for the unknown plant is formed by combining the control law (7.3.12), summarized in Table 7.1, with an adaptive law based on the parametric model (7.4.2) or (7.4.3). The adaptive law generates on-line estimates θa , θb of the coefficient vectors, θa∗ of Rp (s) = sn + θa∗> αn−1 (s) and θb∗ of Zp (s) = θb∗> αn−1 (s) respectively, to form the estimated plant polynomials ˆ p (s, t) = sn + θa> αn−1 (s), Zˆp (s, t) = θb> αn−1 (s) R The estimated plant polynomials are used to compute the estimated conˆ t), Pˆ (s, t) by solving the Diophantine equation troller polynomials L(s, ˆ m·R ˆ p + Pˆ · Zˆp = A∗ LQ

(7.4.21)

ˆ Pˆ pointwise in time or the algebraic equation for L, Sˆl βˆl = αl∗

(7.4.22)

7.4. INDIRECT APPC SCHEMES

475

ˆ p Qm , Zˆp ; βˆl contains the coeffifor βˆl , where Sˆl is the Sylvester matrix of R ∗ ˆ ˆ cients of L, P ; and αl contains the coefficients of A∗ (s) as shown in Table 7.1. The control law in the unknown parameter case is then formed as ˆ m ) 1 up − Pˆ 1 (yp − ym ) up = (Λ − LQ Λ Λ

(7.4.23)

Because different adaptive laws may be picked up from Tables 4.2 to 4.5, a wide class of APPC schemes may be developed. As an example, we present in Table 7.4 the main equations of an APPC scheme that is based on the gradient algorithm of Table 4.2. The implementation of the APPC scheme of Table 7.4 requires that ˆ Pˆ or of the algethe solution of the polynomial equation (7.4.21) for L, braic equation (7.4.22) for βˆl exists at each time. The existence of this ˆ p (s, t)Qm (s), Zˆp (s, t) are coprime at solution is guaranteed provided that R each time t, i.e., the Sylvester matrix Sˆl (t) is nonsingular at each time t. ˆ Pˆ to be uniIn fact for the coefficient vectors l, p of the polynomials L, formly bounded for bounded plant parameter estimates θp , the polynomials ˆ p (s, t)Qm (s), Zˆp (s, t) have to be strongly coprime which implies that their R Sylvester matrix should satisfy | det(Sl (t))| ≥ ν0 > 0 for some constant ν0 at each time t. Such a strong condition cannot be guaranteed by the adaptive law without any additional modifications, giving rise to the so called “stabilizability” or “admissibility” problem to be discussed in Section 7.6. As in the scalar case, the stabilizability problem arises from the fact that the control law is chosen to stabilize the estimated ˆ p (s, t)) at each time. For such a control law plant (characterized by Zˆp (s, t), R to exist, the estimated plant has to satisfy the usual observability, controllability conditions which in this case translate into the equivalent condition of ˆ p (s, t)Qm (s), R ˆ p (s, t) being coprime. The stabilizability problem is one of R the main drawbacks of indirect APPC schemes in general and it is discussed in Section 7.6. In the meantime let us assume that the estimated plant is ˆ p Qm , Zˆp are strongly coprime ∀t ≥ 0 and proceed with stabilizable, i.e., R the analysis of the APPC scheme presented in Table 7.4. ˆ p Qm , Zˆp are Theorem 7.4.1 Assume that the estimated plant polynomials R strongly coprime at each time t. Then all the signals in the closed-loop

476

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

Table 7.4 APPC scheme: polynomial approach. Z (s)

Plant Reference signal

Adaptive law

Calculation

yp = Rpp (s) up Zp (s) = θb∗> αn−1 (s) Rp (s) = sn + θa∗> αn−1 (s) αn−1 (s) = [sn−1 , sn−2 , . . . , s, 1]> , θp∗ = [θb∗> , θa∗> ]> Qm (s)ym = 0 Gradient algorithm from Table 4.2 θ˙p = Γ²φ, Γ = Γ> > 0 ² = (z − θp> φ)/m2 , m2 = 1 + φ> φ α>

(s)

α>

(s)

φ = [ Λn−1 up , − Λn−1 yp ]> p (s) p (s) n z = Λsp (s) yp , θp = [θb> , θa> ]> Zˆp (s, t) = θb> αn−1 (s), Rˆp (s, t) = sn + θa> αn−1 (s) ˆ t) = sn−1 + l> αn−2 (s), Solve for L(s, Pˆ (s, t) = p> αn+q−1 (s) the polynomial equation: ˆ t)·Qm (s)· R ˆ p (s, t) + Pˆ (s, t)· Zˆp (s, t) = A∗ (s) L(s, or solve for βˆl the algebraic equation Sˆl βˆl = αl∗ ˆ p Qm , Zˆp where Sˆl is the Sylverster matrix of R > > > 2(n+q) βˆl = [lq , p ] ∈ R , lq = [0, . . . , 0, 1, l> ]>∈ Rn+q | {z } q

A∗ (s) = s2n+q−1 + α∗> α2n+q−2 (s) αl∗ = [0, . . . , 0, 1, α∗> ]> ∈ R2(n+q) | {z } q

Control law

ˆ m ) 1 up − Pˆ 1 (yp − ym ) up = (Λ − LQ Λ Λ

Design variables

A∗ (s) monic Hurwitz; Λ(s) monic Hurwitz of degree n + q − 1; for simplicity, Λ(s) = Λp (s)Λq (s), where Λp (s), Λq (s) are monic Hurwitz of degree n, q − 1, respectively

7.4. INDIRECT APPC SCHEMES

477

APPC scheme of Table 7.4 are u.b. and the tracking error converges to zero asymptotically with time. The same result holds if we replace the gradient algorithm in Table 7.4 with any other adaptive law from Tables 4.2 and 4.3. Outline of Proof: The proof is completed in the following four steps as in Example 7.4.1: Step 1. Manipulate the estimation error and control law equations to express the plant input up and output yp in terms of the estimation error. This step leads to the following equations: x˙ up yp

= A(t)x + b1 (t)²m2 + b2 y¯m = C1> x + d1 ²m2 + d2 y¯m = C2> x + d3 ²m2 + d4 y¯m

(7.4.24)

where y¯m ∈ L∞ ; A(t), b1 (t) are u.b. because of the boundedness of the estimated plant and controller parameters (which is guaranteed by the adaptive law and the stabilizability assumption); b2 is a constant vector; C1 and C2 are vectors whose elements are u.b.; and d1 to d4 are u.b. scalars. Step 2. Establish the e.s. of the homogeneous part of (7.4.24). The matrix A(t) has stable eigenvalues at each frozen time t that are equal to the roots of A∗ (s) = 0. In addition θ˙p , l,˙ p˙ ∈ L2 (guaranteed by the adaptive law and the stabilizability ˙ assumption), imply that kA(t)k ∈ L2 . Therefore, using Theorem 3.4.11, we conclude that the homogeneous part of (7.4.24) is u.a.s. Step 3. Use the properties of the L2δ norm and B-G Lemma to establish 4 boundedness. Let m2f = 1 + kup k2 + kyp k2 where k · k denotes the L2δ norm. Using the results established in Steps 1 and 2 and the normalizing properties of mf , we show that (7.4.25) m2f ≤ ck²mmf k2 + c which implies that

Z m2f

≤c 0

t

e−δ(t−τ ) ²2 m2 m2f dτ + c

Because ²m ∈ L2 , the boundedness of mf follows by applying the B-G lemma. Using the boundedness of mf , we can establish the boundedness of all signals in the closed-loop plant. Step 4. Establish that the tracking error e1 converges to zero. The convergence of e1 to zero follows by using the control and estimation error equations to express e1 as the output of proper stable LTI systems whose inputs are in L2 ∩ L∞ . The details of the proof of Theorem 7.4.1 are given in Section 7.7. 2

478

7.4.3

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

APPC Schemes: State-Variable Approach

As in Section 7.4.2, let us start with a scalar example to illustrate the design and analysis of an APPC scheme formed by combining the control law of Section 7.3.3 developed for the case of known plant parameters with an adaptive law. Example 7.4.2 We consider the same plant as in Example 7.3.2, i.e., yp =

b up s+a

(7.4.26)

where a and b are unknown constants with b 6= 0 and up is to be chosen so that the poles of the closed-loop plant are placed at the roots of A∗ (s) = (s + 1)2 = 0 and yp tracks the reference signal ym = 1. As we have shown in Example 7.3.2, if a, b are known, the following control law can be used to meet the control objective: · ¸ · ¸ −a 1 1 eˆ˙ = eˆ + b¯ up − Ko ([1 0]ˆ e − e1 ) 0 0 1 s+1 u ¯p = −Kc eˆ, up = u ¯p (7.4.27) s where Ko , Kc are calculated by solving the equations det(sI − A + BKc ) = (s + 1)2 det(sI − A + Ko C > ) = (s + 5)2 where

· A=

−a 1 0 0

i.e., Kc =

¸

· , B=

1 1

¸ b, C > = [1, 0]

1 [1 − a, 1], Ko = [10 − a, 25]> b

(7.4.28)

The APPC scheme for the plant (7.4.26) with unknown a and b may be formed by replacing the unknown parameters a and b in (7.4.27) and (7.4.28) with their on-line estimates a ˆ and ˆb generated by an adaptive law as follows: Adaptive Law The adaptive law uses the measurements of the plant input up and output yp to generate a ˆ, ˆb. It is therefore independent of the choice of the control law and the same adaptive law as the one used in Example 7.4.1 can be employed, i.e., θ˙p

= Pr{Γ²φ}, Γ = Γ> > 0

7.4. INDIRECT APPC SCHEMES z − θp> φ s , m2 = 1 + φ> φ, z = yp m2 s+λ ¸ · ¸ · ˆb 1 up , φ= a ˆ s + λ −yp

² = θp

479

=

(7.4.29)

where λ > 0 is a design constant and Pr {·} is the projection operator as defined in Example 7.4.1 that guarantees ˆb(t)| ≥ b0 > 0 ∀t ≥ 0. State Observer

ˆ e + B(t)¯ ˆ up − K ˆ o (C > eˆ − e1 ) eˆ˙ = A(t)ˆ

where

· Aˆ =

−ˆ a 1 0 0

¸

· ˆ= , B

1 1

(7.4.30)

¸ ˆb, C > = [1, 0]

ˆ c, K ˆ o by solving Calculation of Controller Parameters Calculate K ˆK ˆ c ) = A∗c (s) = (s + 1)2 , det(sI − Aˆ + K ˆ o C > ) = (s + 5)2 det(sI − Aˆ + B for each frozen time t which gives ˆ c = 1 [1 − a ˆ o = [10 − a K ˆ, 1], K ˆ, 25]> ˆb

(7.4.31)

Control Law:

ˆ c eˆ, up = s + 1 u u ¯p = −K ¯p (7.4.32) s ˆ c exists for any monic Hurwitz The solution for the controller parameter vector K ∗ ˆ ˆ polynomial Ac (s) of degree 2 provided (A, B) is stabilizable and A∗c (s) contains the ˆ B) ˆ loses uncontrollable eigenvalues of Aˆ as roots. For the example considered, (A, ˆ ˆ its controllability when b = 0. It also loses its controllability when b 6= 0 and a ˆ = 1. ˆ B), ˆ even though uncontrollable, is stabilizable and In this last case, however, (A, the uncontrollable eigenvalue is at s = −1, which is a zero of A∗c (s) = (s + 1)2 . ˆ c exists for all a Therefore, as it is also clear from (7.4.31), K ˆ, ˆb provided ˆb 6= 0. Because the projection operator in (7.4.29) guarantees as in Example 7.4.1 that ˆ c follows from a |ˆb(t)| ≥ b0 > 0 ∀t ≥ 0, the existence and boundedness of K ˆ, ˆb ∈ L∞ . Analysis Step 1. Develop the state error equations for the closed-loop APPC scheme. The state error equations for the closed-loop APPC scheme include the tracking error equation and the observer equation. The tracking error equation e1 =

b(s + 1) u ¯p , s(s + a)

u ¯p =

s up s+1

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CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

is expressed in the state space form · ¸ · ¸ −a 1 1 e˙ = e+ b¯ up 0 0 1 e1

(7.4.33)

= C > e = [1, 0]e

Let eo = e − eˆ be the observation error. Then from (7.4.30) and (7.4.33) we obtain the state equations · ¸ −ˆ a + 10 eˆ˙ = Ac (t)ˆ e+ C > eo 25 · ¸ · ¸ · ¸ −10 1 1 1 ˜ b¯ up e˙ o = eo + a ˜ e1 − −25 0 0 1 ˆ c eˆ u ¯ p = −K (7.4.34) 4

4

4

ˆ −B ˆK ˆ c, a where Ac (t) = A(t) ˜=a ˆ − a, ˜b = ˆb − b. The plant output is related to e0 , eˆ as follows: yp = e1 + ym = C > (e0 + eˆ) + ym (7.4.35) The relationship between up and e0 , eˆ may be developed as follows: The coprimeness of b, s(s + a) implies the existence of polynomials X(s), Y (s) of degree 1 and with X(s) monic such that s(s + a)X(s) + b(s + 1)Y (s) = A∗ (s)

(7.4.36)

where A∗ (s) = (s+1)a∗ (s) and a∗ (s) is any monic polynomial of degree 2. Choosing . Equation (7.4.36) a∗ (s) = (s + 1)2 , we obtain X(s) = s + 1 and Y (s) = (2−a)s+1 b may be written as s(s + a) (2 − a)s + 1 + =1 (7.4.37) (s + 1)2 (s + 1)2 which implies that up = Using up =

s+a b yp , sup

s(s + a) (2 − a)s + 1 up + up (s + 1)2 (s + 1)2

ˆ c eˆ, we have = (s + 1)¯ up and u ¯ p = −K

up = −

s+a ˆ (2 − a)s + 1 (s + a) Kc eˆ + yp s+1 (s + 1)2 b

(7.4.38)

Equations (7.4.34), (7.4.35), and (7.4.38) describe the stability properties of the closed-loop APPC scheme. Step 2. Establish the e.s. of the homogeneous part of (7.4.34). The homogeneous part of (7.4.34) is considered to be the part with the input a ˜e1 , ˜b¯ up set equal

7.4. INDIRECT APPC SCHEMES

481

to zero. The e.s. of the homogeneous part · of (7.4.34) ¸ can be established by showing −10 1 that Ac (t) is a u.a.s matrix and Ao = is a stable matrix. Because Ao −25 0 ˆ − B(t) ˆ K ˆ c (t) is u.a.s. is stable by design, it remains to show that Ac (t) = A(t) The projection used in the adaptive law guarantees as in Example 7.4.1 that ˆ c , Ac are |ˆb(t)| ≥ b0 > 0, ∀t ≥ 0. Because ˆb, a ˆ ∈ L∞ it follows that the elements of K u.b. Furthermore, at each time t, λ(Ac (t)) = −1, −1, i.e., Ac (t) is a stable matrix ˙ ˆ˙ c | ∈ L2 . Hence, at each frozen time t. From (7.4.31) and a ˆ˙ , ˆb ∈ L2 , we have |K kA˙ c (t)k ∈ L2 and the u.a.s of Ac (t) follows by applying Theorem 3.4.11. In the rest of the proof, we exploit the u.a.s of the homogeneous part of (7.4.34) and the relationships of the inputs a ˜e1 , ˜b¯ up with the properties of the adaptive law in an effort to first establish signal boundedness and then convergence of the tracking error to zero. In the analysis we employ the L2δ norm k(·)t k2δ which for clarity of presentation we denote by k · k. Step 3. Use the properties of the L2δ norm and B-G Lemma to establish boundedness. Applying Lemmas 3.3.2 and 3.3.3 to (7.4.34), (7.4.35), and (7.4.38), we obtain kˆ ek kyp k kup k

≤ ≤ ≤

ckC > eo k ckC > eo k + ckˆ ek + c ckˆ ek + ckyp k + c

for some δ > 0 where c ≥ 0 denotes any finite constant, which imply that kyp k ≤ kup k ≤

ckC > eo k + c ckC > eo k + c

Therefore, the fictitious normalizing signal 4

m2f = 1 + kyp k2 + kup k2 ≤ ckC > eo k2 + c

(7.4.39)

We now need to find an upper bound for kC > eo k, which is a function of the L2 ˙ signals ²m, a ˆ˙ , ˆb. From equation (7.4.34) we write C > eo =

s s+1 ˜ s a ˜e1 − b up 2 (s + 5) (s + 5)2 s + 1

Applying the Swapping Lemma A.1 (see Appendix A) to the above equation and using the fact that se1 = syp we obtain C > eo = a ˜

s s ˙ yp − ˜b up + Wc1 (Wb1 e1 )a ˆ˙ − Wc2 (Wb2 up )ˆb 2 2 (s + 5) (s + 5)

(7.4.40)

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CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

where the elements of Wci (s), Wbi (s) are strictly proper transfer functions with poles at −1, −5. To relate the first two terms on the right-hand side of (7.4.40) with ²m2 , we use (7.4.29) and z = θp∗> φ to write ²m2 = z − θp> φ = −θ˜p> φ = a ˜ We filter both sides of (7.4.41) with to obtain

s(s+λ) (s+5)2

1 1 yp − ˜b up s+λ s+λ

(7.4.41)

and then apply the Swapping Lemma A.1

n o s(s + λ) 2 s s ˜b ˙ − (Wb up )ˆb˙ ²m = a ˜ y − u + W (W y ) a ˆ p p c b p (s + 5)2 (s + 5)2 (s + 5)2

(7.4.42)

where the elements of Wc (s), Wb (s) are strictly proper transfer functions with poles at −5. Using (7.4.42) in (7.4.40), we have that C > eo =

s(s + λ) 2 ˙ ²m + G(s, a ˆ˙ , ˆb) (s + 5)2

(7.4.43)

where n o ˙ ˙ ˙ G(s, a ˆ˙ , ˆb) = Wc1 (Wb1 e1 )a ˆ˙ − Wc2 (Wb2 up )ˆb − Wc (Wb yp )a ˆ˙ − (Wb up )ˆb Using Lemma 3.3.2 we can establish that φ/mf , m/mf , Wb1 e1 /mf , Wb2 up /mf , Wb yp /mf , Wb up /mf ∈ L∞ . Using the same lemma we have from (7.4.43) that ˙ kC > eo k ≤ ck²mmf k + ckmf a ˆ˙ k + ckmf ˆbk

(7.4.44)

Combining (7.4.39), (7.4.44), we have m2f ≤ ckgmf k2 + c or

Z m2f ≤ c

0

t

(7.4.45)

e−δ(t−τ ) g 2 (τ )m2f (τ )dτ + c

2 ˙ ˙ where g 2 (τ ) = ²2 m2 + a ˆ˙ + ˆb2 . Because ²m, a ˆ˙ , ˆb ∈ L2 , it follows that g ∈ L2 and therefore by applying B-G Lemma to (7.4.45), we obtain mf ∈ L∞ . The boundedness of mf implies that φ, m, Wb1 e1 , Wb2 up , Wb yp , Wb up ∈ L∞ . Because T T ˙ > a ˆ˙ , ˆb, ²m2 ∈ L∞ L2 , it follows that CT eo ∈ L∞ L2 by applying Corollary 3.3.1 to (7.4.43). Now by using C > eo ∈ L∞ L2 in (7.4.34), it follows T from the stability of Ac (t) and Lemma 3.3.3 or Corollary 3.3.1 that eˆ ∈ L∞ L2 and T eˆ(t) → 0 as t → ∞. Hence, from (7.4.35), we have yp ∈ L∞ and e1 ∈ L∞ L2 and from (7.4.38) that up ∈ L∞ .

7.4. INDIRECT APPC SCHEMES

483

Step 4. Convergence of the trackingTerror to zero. Because e˙ 1 ∈ L∞ (due to e˙ 1 = y˙ p and y˙ p ∈ L∞ ) and e1 ∈ L∞ L2 , it follows from Lemma 3.2.5 that e1 (t) → 0 as t → ∞. We can continue the analysis and establish that ², ²m → 0 as t → ∞, which ˆ˙ c → 0 as t → ∞. implies that θ˙p , K ˆ c to Kc = 1 [1 − a, 1] cannot be The convergence of θp to θp∗ = [b, a]> and of K b guaranteed, however, unless the signal vector φ is PE. For φ to be PE, the reference signal ym has to be sufficiently rich of order 2. Because ym = 1 is sufficiently rich of order 1, φ ∈ R2 is not PE. 5

General Case Let us now extend the results of Example 7.4.2 to the nth-order plant (7.3.1). We design an APPC scheme for the plant (7.3.1) by combining the state feedback control law of Section 7.3.3 summarized in Table 7.2 with any appropriate adaptive law from Tables 4.2 to 4.5 based on the plant parametric model (7.4.2) or (7.4.3). ˆ p (s, t), Zˆp (s, t) of the The adaptive law generates the on-line estimates R unknown plant polynomials Rp (s), Zp (s), respectively. These estimates are ˆ of the unknown matrices A and B, used to generate the estimates Aˆ and B respectively, that are used to calculate the controller parameters and form the observer equation. Without loss of generality, we concentrate on parametric model (7.4.2) and select the gradient algorithm given in Table 4.2. The APPC scheme formed is summarized in Table 7.5. The algebraic equation for calculating the controller parameter vector ˆ ˆ c (t) at each time t provided the Kc in Table 7.5 has a finite solution for K ˆ B) ˆ is controllable which is equivalent to Zˆp (s, t)Q1 (s), R ˆ p (s, t)Qm (s) pair (A, ˆ being coprime at each time t. In fact, for Kc (t) to be uniformly bounded, ˆ B) ˆ has to be strongly controllable, i.e., the absolute value of the de(A, ˆ p (s, t)Qm (s) has to be terminant of the Sylvester matrix of Zˆp (s, t)Q1 (s), R greater than some constant ν0 > 0 for all t ≥ 0. This strong controllability condition may be relaxed by choosing Q1 (s) to be a factor of the desired closed-loop Hurwitz polynomial A∗c (s) as indicated in Table 7.5. By doing ˆ p (s, t) to have common factors without affecting the so, we allow Q1 (s), R ˆ c , because such common factors are solvability of the algebraic equation for K

484

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

Table 7.5 APPC scheme: state feedback law Z (s)

Plant

Reference signal

yp = Rpp (s) up Zp (s) = θb∗> αn−1 (s), Rp (s) = sn + θa∗> αn−1 (s) £ ¤> αn−1 (s) = sn−1 , sn−2 , . . . s, 1 Qm (s)ym = 0 Gradient algorithm based on z = θp∗> φ θ˙p = Γ²φ, Γ = Γ> > 0

z−θp> φ , m2 = 1 + φ> φ 2 · m> ¸> n α (s) α> n−1 (s) φ = Λn−1 u , − y z = Λsp (s) yp p Λp (s) p , p (s) h i> θp = θb> , θa> ˆ p (s, t) = sn +θa>(t)αn−1(s) Zˆp (s, t) = θb> (t)αn−1 (s), R

²=

Adaptive law

State observer

Calculation of controller parameters Control law

Design variables

>e ˆe + B ˆu ˆ eˆ˙ =Aˆ ˆ − e1 ), eˆ ∈ Rn+q ¯¯p − Ko (t)(C  ¯ I ¯ n+q−1  ˆ −θ1 (t) ¯¯ − − −−  ˆ 2 (t), C >=[1, 0, . . . , 0] A=  , B=θ ¯ ¯ 0 n+q ˆ p Qm −sn+q θ1∈R is the coefficient vector of R θ2 ∈ Rn+q is the coefficient vector of Zˆp Q1 ˆ o = α∗ − θ1 , and α∗ is the coefficient vector of K A∗o (s) − sn+q

ˆ c pointwise in time the equation Solve for K ˆK ˆ c ) = A∗c (s) det(sI − Aˆ + B ˆ c (t)ˆ u ¯ p = −K e, up = QQm1 u ¯p Qm (s) monic of degree q with nonrepeated roots on the jω-axis; Q1 (s) monic Hurwitz of degree q; A∗0(s) monic Hurwitz of degree n+q; A∗c(s) monic Hurwitz of degree n + q and with Q1 (s) as a factor; Λp (s) monic Hurwitz of degree n

7.4. INDIRECT APPC SCHEMES

485

also included in A∗c (s). Therefore the condition that guarantees the existence ˆ c is that Zˆp (s, t), R ˆ p (s, t)Qm (s) are strongly and uniform boundedness of K coprime at each time t. As we mentioned earlier, such a condition cannot be guaranteed by any one of the adaptive laws developed in Chapter 4 without additional modifications, thus giving rise to the so-called stabilizability or admissibility problem to be discussed in Section 7.6. In this section, we ˆ p (s, t)Qm (s) are strongly coprime at assume that the polynomials Zˆp (s, t), R each time t and proceed with the analysis of the APPC scheme of Table 7.5. We relax this assumption in Section 7.6 where we modify the APPC schemes to handle the possible loss of stabilizability of the estimated plant. ˆ p Qm are strongly coTheorem 7.4.2 Assume that the polynomials Zˆp , R prime at each time t. Then all the signals in the closed-loop APPC scheme of Table 7.5 are uniformly bounded and the tracking error e1 converges to zero asymptotically with time. The same result holds if we replace the gradient algorithm with any other adaptive law from Tables 4.2 to 4.4 that is based on the plant parametric model (7.4.2) or (7.4.3). Outline of Proof Step 1. Develop the state error equations for the closed-loop APPC scheme, i.e., eˆ˙ = e˙ o = yp = up

=

ˆ o C > eo Ac (t)ˆ e+K Ao eo + θ˜1 e1 − θ˜2 u ¯p C > eo + C > eˆ + ym ˆ c (t)ˆ W1 (s)K e + W2 (s)yp

u ¯p

=

ˆ c eˆ −K

(7.4.46)

4

where eo = e − eˆ is the observation error, Ao is a constant stable matrix, W1 (s) and ˆ B ˆK ˆ c. W2 (s) are strictly proper transfer functions with stable poles, and Ac (t) = A− ˆ c is Step 2. Establish e.s. for the homogeneous part of (7.4.46). The gain K chosen so that the eigenvalues of Ac (t) at each time t are equal to the roots of the ˆ B ˆ ∈ L∞ (guaranteed by the adaptive law) Hurwitz polynomial A∗c (s). Because A, ˆ p Qm are strongly coprime (by assumption), we conclude that (A, ˆ B) ˆ is and Zˆp , R ˆ c , Ac ∈ L∞ . Using θ˙a , θ˙b ∈ L2 , guaranteed by stabilizable in a strong sense and K ˆ˙ c , A˙ c ∈ L2 . Therefore, applying Theorem 3.4.11, we the adaptive law, we have K have that Ac (t) is a u.a.s. matrix. Because Ao is a constant stable matrix, the e.s. of the homogeneous part of (7.4.46) follows.

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CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

Step 3. Use the properties of the L2δ norm and the B-G Lemma to establish signal boundedness. We use the properties of the L2δ norm and equation (7.4.46) to establish the inequality m2f ≤ ckgmf k2 + c 4 where g 2 = ²2 m2 + |θ˙a |2 + |θ˙b |2 and m2f = 1 + kup k2 + kyp k2 is the fictitious normalizing signal. Because g∈ L2 , it follows that mf ∈ L∞ by applying the B-G Lemma. Using mf ∈ L∞ , we establish the boundedness of all signals in the closedloop plant.

Step 4. Establish the convergence of the tracking error e1 to zero. This is done by following the same procedure as in Example 7.4.2. 2

The details of the proof of Theorem 7.4.2 are given in Section 7.7.

7.4.4

Adaptive Linear Quadratic Control (ALQC)

The linear quadratic (LQ) controller developed in Section 7.3.4 can be made adaptive and used to meet the control objective when the plant parameters are unknown. This is achieved by combining the LQ control law (7.3.28) to (7.3.32) with an adaptive law based on the plant parametric model (7.4.2) or (7.4.3). We demonstrate the design and analysis of ALQ controllers using the following examples: Example 7.4.3 We consider the same plant and control objective as in Example 7.3.3, given by x˙ = yp =

−ax + bup x

(7.4.47)

where the plant input up is to be chosen to stabilize the plant and regulate yp to zero. In contrast to Example 7.3.3, the parameters a and b are unknown constants. The control law up = − λ1 bpyp in Example 7.3.3 is modified by replacing the unknown plant parameters a, b with their on-line estimates a ˆ and ˆb generated by the same adaptive law used in Example 7.4.2, as follows: Adaptive Law θ˙p ²

= Γ²φ, =

Γ = Γ> > 0

z − θp> φ , m2

m2 = 1 + φ> φ,

z=

s yp s + λ0

7.4. INDIRECT APPC SCHEMES · θp

=

ˆb a ˆ

¸ ,

φ=

487

1 s + λ0

·

up −yp

¸

where λ0 > 0 is a design constant. Control Law

1 up = − ˆb(t)p(t)yp λ

(7.4.48)

Riccati Equation Solve the equation −2ˆ a(t)p(t) −

p2 (t)ˆb2 (t) +1=0 λ

at each time t for p(t) > 0, i.e., p(t) =

−λˆ a+

p λ2 a ˆ2 + ˆb2 λ >0 ˆb2

(7.4.49)

As in the previous examples, for the solution p(t) in (7.4.49) to be finite, the estimate ˆb should not cross zero. In fact, for p(t) to be uniformly bounded, ˆb(t) should satisfy |ˆb(t)| ≥ b0 > 0, ∀t ≥ 0 for some constant b0 that satisfies |b| ≥ b0 . Using the knowledge of b0 and sgn(b), the adaptive law for ˆb can be modified as before to guarantee |ˆb(t)| ≥ b0 , ∀t ≥ 0 and at the same time retain the properties that θp ∈ L∞ and ², ²m, θ˙p ∈ L2 ∩ L∞ . The condition |ˆb(t)| ≥ b0 implies that the estimated plant, characterized by the parameters a ˆ, ˆb, is strongly controllable at each time t, a condition required for the solution p(t) > 0 of the Riccati equation to exist and be uniformly bounded. Analysis For this first-order regulation problem, the analysis is relatively simple and can be accomplished in the following four steps: Step 1. Develop the closed-loop error equation. The closed-loop plant can be written as ˆb2 p x˙ = −(ˆ a+ )x + a ˜x − ˜bup (7.4.50) λ by adding and subtracting a ˆx − ˆbup and using up = −ˆbpx/λ. The inputs a ˜x, ˜bup are 4 4 due to the parameter errors a ˜=a ˆ − a, ˜b = ˆb − b. Step 2. Establish the e.s. of the homogeneous part of (7.4.50). The eigenvalue of the homogeneous part of (7.4.50) is −(ˆ a+

ˆb2 p ) λ

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CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

which is guaranteed to be negative by the choice of p(t) given by (7.4.49), i.e., s ˆb2 p ˆb2 b0 −(ˆ a+ ˆ2 + )=− a ≤ −√ < 0 λ λ λ provided, of course, the adaptive law is modified by using projection to guarantee |ˆb(t)| ≥ b0 , ∀t ≥ 0. Hence, the homogeneous part of (7.4.50) is e.s. Step 3. Use the properties of the L2δ norm and B-G Lemma to establish boundedness. The properties of the input a ˜x − ˜bup in (7.4.50) depend on the properties of the adaptive law that generates a ˜ and ˜b. The first task in this step is to establish ˙ the smallness of the input a ˜x − ˜bup by relating it with the signals a ˜˙ , ˜b, and εm that are guaranteed by the adaptive law to be in L2 . We start with the estimation error equation εm2 = z − θp> φ = −θ˜p> φ = a ˜

1 1 x − ˜b up s + λ0 s + λ0

(7.4.51)

Operating with (s + λ0 ) on both sides of (7.4.51) and using the property of differ4

d entiation, i.e., sxy = xy˙ + xy ˙ where s = dt is treated as the differential operator, we obtain 1 ˙ 1 (s + λ0 )εm2 = a ˜x − ˜bup + a ˜˙ x − ˜b up (7.4.52) s + λ0 s + λ0

Therefore, a ˜x − ˜bup = (s + λ0 )εm2 − a ˜˙

1 ˙ 1 x + ˜b up s + λ0 s + λ0

which we substitute in (7.4.50) to obtain x˙ = −(ˆ a+

ˆb2 p 1 ˙ 1 )x + (s + λ0 )εm2 − a x + ˜b up ˜˙ λ s + λ0 s + λ0

(7.4.53)

4

If we define e¯ = x − εm2 , (7.4.53) becomes ˆb2 p ˆb2 p 1 ˙ 1 )¯ e + (λ0 − a ˆ− )εm2 − a ˜˙ x + ˜b up (7.4.54) λ λ s + λ0 s + λ0

e¯˙ =

−(ˆ a+

x =

e¯ + ²m2

Equation (7.4.54) has a homogeneous part that is e.s. and an input that is small in ˙ some sense because of εm, a ˜˙ , ˜b ∈ L2 . Let us now use the properties of the L2δ norm, which for simplicity is denoted by k · k to analyze (7.4.54). The fictitious normalizing signal mf satisfies 4

m2f = 1 + kyp k2 + kup k2 ≤ 1 + ckxk2

(7.4.55)

7.4. INDIRECT APPC SCHEMES

489

for some δ > 0 because of the control law chosen and the fact that ˆb, p ∈ L∞ . Because x = e¯ + εm2 , we have kxk ≤ k¯ ek + kεm2 k, which we use in (7.4.55) to obtain m2f ≤ 1 + ck¯ ek2 + ck²m2 k2 (7.4.56) From (7.4.54), we have ˙ k¯ ek2 ≤ ck²m2 k2 + cka ˜˙ x ¯k2 + ck˜b¯ up k2

(7.4.57)

1 1 where x ¯ = s+λ x, u ¯p = s+λ up . Using the properties of the L2δ norm, it can be 0 0 shown that mf bounds from above m, x ¯, u ¯p and therefore it follows from (7.4.56), (7.4.57) that ˙ m2f ≤ 1 + ck²mmf k2 + cka ˜˙ mf k2 + ck˜bmf k2 (7.4.58)

which implies that Z m2f ≤ 1 + c

t 0

e−δ(t−τ ) g 2 (τ )m2f (τ )dτ

2 4 ˙2 ˙ where g 2 = ε2 m2 + a ˜˙ + ˜b . Since εm, a ˜˙ , ˜b ∈ L2 imply that g ∈ L2 , the boundedness of mf follows by applying the B-G Lemma. Now mf ∈ L∞ implies that m, x ¯, u ¯p and, therefore, φ ∈ L∞ . Using (7.4.54), ˙ 2 ˜ ˙ and the fact that a ˜, b, εm , x ¯ = (1/(s + λ0 )) x, u ¯p = (1/(s + λ0 )) up ∈ L∞ , we have e¯ ∈ L∞ , which implies that x = yp ∈ L∞ , and therefore up and all signals in the closed loop plant are bounded.

Step 4. Establish that x = yp → 0 as t → ∞. We proceed as follows: Using (7.4.54), we establish that e¯ ∈ L2 , which together with ²m2 ∈ L2 imply that x = e¯ + ²m2 ∈ L2 . Because (7.4.53) implies that x˙ ∈ L∞ , it follows from Lemma 3.2.5 that x(t) → 0 as t → ∞. The analysis of the ALQ controller presented above is simplified by the fact that the full state is available for measurement; therefore, no state observer is necessary. Furthermore, the u.a.s. of the homogeneous part of (7.4.50) is established by simply √ showing that the time–varying scalar a ˆ + ˆb2 p/λ ≥ b0 / λ > 0, ∀t ≥ 0, i.e., that the closed–loop eigenvalue is stable at each time t, which in the scalar case implies u.a.s. 5

In the following example, we consider the tracking problem for the same scalar plant given by (7.4.47). In this case the analysis requires some additional arguments due to the use of a state observer and higher dimensionality. Example 7.4.4 Let us consider the same plant as in Example 7.4.3 but with the following control objective: Choose up to stabilize the plant and force yp to follow

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CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

the constant reference signal ym (t) = 1. This is the same control problem we solved in Example 7.3.4 under the assumption that the plant parameters a and b are known exactly. The control law when a, b are known given in Example 7.3.4 is summarized below: State Observer

eˆ˙ = Aˆ e + Bu ¯p − Ko (C > eˆ − e1 ), e1 = yp − ym u ¯p = −λ−1 [b, b]P eˆ, up =

Control Law

s+1 u ¯p s

Riccati Equation A> P + P A − P BB > P λ−1 + CC > = 0 · ¸ · ¸ −a 1 1 where A = , B= b, C > = [1, 0], Ko = [10 − a, 25]> 0 0 1 In this example, we assume that a and b are unknown constants and use the certainty equivalence approach to replace the unknown a, b with their estimates a ˆ, ˆb generated by an adaptive law as follows: State Observer

ˆ e+B ˆu ˆ o (t) ([1 0]ˆ eˆ˙ = A(t)ˆ ¯p − K e − e1 ) · ¸ · ¸ · ¸ −ˆ a 1 1 10 − a ˆ ˆ ˆ ˆ ˆ A= , B=b , Ko = 0 0 1 25

(7.4.59)

Control Law

ˆb s+1 u ¯p = − [1 1]P eˆ, up = u ¯p (7.4.60) λ s Riccati Equation Solve for P (t) = P > (t) > 0 at each time t the equation ˆB ˆ> B Aˆ> P + P Aˆ − P P + CC > = 0 , C > = [1, 0] (7.4.61) λ The estimates a ˆ(t) and ˆb(t) are generated by the same adaptive law as in Example 7.4.3. ˆ B) ˆ has to be For the solution P = P > > 0 of (7.4.61) to exist, the pair (A, ˆ b(s+1) ˆ B, ˆ C) is the realization of stabilizable. Because (A, , the stabilizability of (s+ˆ a)s

ˆ B) ˆ is guaranteed provided ˆb 6= 0 (note that for a ˆ B) ˆ is (A, ˆ = 1, the pair (A, no longer controllable but it is still stabilizable). In fact for P (t) to be uniformly bounded, we require |ˆb(t)| ≥ b0 > 0, for some constant b0 , which is a lower bound for |b|. As in the previous examples, the adaptive law for ˆb can be modified to guarantee |ˆb(t)| ≥ b0 , ∀t ≥ 0 by assuming that b0 and sgn(b) are known a priori. Analysis The analysis is very similar to that given in Example 7.4.2. The tracking error equation is given by · ¸ · ¸ −a 1 1 e˙ = e+ b¯ up , e1 = [1, 0]e 0 0 1

7.4. INDIRECT APPC SCHEMES

491

If we define eo = e − eˆ to be the observation error and use the control law (7.4.60) in (7.4.59) , we obtain the same error equation as in Example 7.4.2, i.e., · eˆ˙ = Ac (t)ˆ e+ · e˙ o =

−10 −25

1 0

−ˆ a + 10 25

¸

· eo +

1 0

¸ C > eo

¸

· a ˜e1 −

1 1

¸ ˜b¯ up

ˆ ˆB ˆ > P/λ and yp , up are related to eo , eˆ through the equations where Ac (t) = A(t)− B yp = C > e0 + C > eˆ + ym up = −

ˆ >P s+aB (2 − a)s + 1 s + a eˆ + yp s+1 λ (s + 1)2 b

If we establish the u.a.s. of Ac (t), then the rest of the analysis is exactly the same as that for Example 7.4.2. Using the results of Section 7.3.4, we can establish that the matrix Ac (t) at each frozen time t has all its eigenvalues in the open left half s-plane. Furthermore, 2 ˙ ˆ˙ ˆ ˆ 2||B(t)|| ||B(t)|| ||P (t)|| ||B(t)|| ||P (t)|| ˙ˆ ||A˙ c (t)|| ≤ ||A(t)|| + + λ λ

˙ˆ ˙ ˆ˙ where ||A(t)||, ||B(t)|| ∈ L2 due to a ˜˙ , ˜b ∈ L2 guaranteed by the adaptive law. By taking the first-order derivative on each side of (7.4.61), P˙ can be shown to satisfy ˙ P˙ Ac + A> c P = −Q

(7.4.62)

where > ˆ˙ B ˆ> ˆB ˆ˙ B B ˙ˆ> ˙ˆ Q(t) = A P + P A − P P −P P λ λ

ˆ B, ˆ and P , (7.4.62) is a Lyapunov equation and its solution P˙ For a given A, exists and is continuous with respect to Q, i.e., kP˙ (t)k ≤ ckQ(t)k for some constant ˙ˆ ˆ˙ ˆ B, ˆ P ∈ L∞ , we have ||P˙ (t)|| ∈ L2 c ≥ 0. Because of kA(t)k, kB(t)k ∈ L2 and A, ˙ and, thus. Ac (t) ∈ L2 . Because Ac (t) is a stable matrix at each frozen time t, we can apply Theorem 3.4.11 to conclude that Ac is u.a.s. The rest of the analysis follows by using exactly the same steps as in the analysis of Example 7.4.2 and is, therefore, omitted. 5

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CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

General Case Following the same procedure as in Examples 7.4.3 and 7.4.4, we can design a wide class of ALQ control schemes for the nth-order plant (7.3.1) by combining the LQ control law of Section 7.3.4 with adaptive laws based on the parametric model (7.4.2) or (7.4.3) from Tables 4.2 to 4.5. Table 7.6 gives such an ALQ scheme based on a gradient algorithm for the nth-order plant (7.3.1). As with the previous APPC schemes, the ALQ scheme depends on the solvability of the algebraic Riccati equation. The Riccati equation is solved ˆ B ˆ of the plant parameters. for each time t by using the on-line estimates A, > ˆ B) ˆ has to be For the solution P (t) = P (t) > 0 to exist, the pair (A, ˆ p (s, t)Qm (s) stabilizable at each time t. This implies that the polynomials R ˆ and Zp (s, t)Q1 (s) should not have any common unstable zeros at each frozen ˆ B) ˆ to be time t. Because Q1 (s) is Hurwitz, a sufficient condition for (A, ˆ p (s, t)Qm (s), Zˆp (s, t) are coprime at stabilizable is that the polynomials R each time t. For P (t) to be uniformly bounded, however, we will require ˆ p (s, t)Qm (s), Zˆp (s, t) to be strongly coprime at each time t. R In contrast to the simple examples considered, the modification of the ˆ p Qm , Zˆp without adaptive law to guarantee the strong coprimeness of R the use of additional a priori information about the unknown plant is not clear. This problem known as the stabilizability problem in indirect APPC is addressed in Section 7.6 . In the meantime, let us assume that the stabilizability of the estimated plant is guaranteed and proceed with the following theorem that states the stability properties of the ALQ control scheme given in Table 7.6. ˆ p (s, t)Qm (s), Zˆp (s, t) are Theorem 7.4.3 Assume that the polynomials R strongly coprime at each time t. Then the ALQ control scheme of Table 7.6 guarantees that all signals in the closed–loop plant are bounded and the tracking error e1 converges to zero as t → ∞. The same result holds if we replace the gradient algorithm in Table 7.6 with any other adaptive law from Tables 4.2 to 4.4 based on the plant parametric model (7.4.2) or (7.4.3). Proof The proof is almost identical to that of Theorem 7.4.2, except for some minor details. The same error equations as in the proof of Theorem 7.4.2 that relate eˆ and the observation error eo = e − eˆ with the plant input and output also

7.4. INDIRECT APPC SCHEMES

493

Table 7.6 Adaptive linear quadratic control scheme Plant

Z (s)

yp = Rpp (s) up Zp (s) = θb∗> αn−1 (s), Rp (s) = sn + θa∗> αn−1 (s) αn−1 (s) = [sn−1 , sn−2 , . . . , s, 1]>

Reference signal

Qm (s)ym = 0

Adaptive law

Same gradient algorithm as in Table 7.5 to generate ˆ p (s, t) = sn + θ> (t)αn−1 (s) Zˆp (s, t) = θb> (t)αn−1 (s), R a

State observer

>e ˆ e+B ˆu ˆ eˆ˙ = A(t)ˆ ˆ − e1 ) ¯ ¯p − Ko (t)(C   ¯ I ¯ n+q−1 ¯  ˆ = ˆ = θ2 (t) A(t) −θ1 ¯ − − −−  , B(t) ¯ ¯ 0 ∗ ˆ Ko (t) = a − θ1 , C = [1, 0, . . . , 0]> ∈ Rn+q ˆ p (s, t)Qm (s)−sn+q θ1 is the coefficient vector of R θ2 is the coefficient vector of Zˆp (s, t)Q1 (s) a∗ is the coefficient vector of A∗o (s) − sn+q θ1 , θ2 , a∗ ∈ Rn+q

Riccati equation

Solve for P (t) = P > (t) > 0 the equation ˆB ˆ > P + CC > = 0 Aˆ> P + P Aˆ − λ1 P B

Control law

ˆ > P eˆ, up = u ¯p = − λ1 B

Design variables

Q1 (s) ¯p Qm (s) u

Qm (s) is a monic polynomial of degree q with nonrepeated roots on the jω axis; A∗o (s) is a monic Hurwitz polynomial of degree n + q with relatively fast zero; λ > 0 as in Table 7.3; Q1 (s) is a monic Hurwitz polynomial of degree q.

494

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

hold here, i.e.,

ˆ o C > eo eˆ˙ = Ac (t)ˆ e+K ˆ c eˆ e˙ o = Ao eo + θ˜1 e1 + θ˜2 K yp = C > eo + C > eˆ + ym ˆ c (t)ˆ up = W1 (s)K e + W2 (s)yp

ˆK ˆ c , we have K ˆc = B ˆ > (t)P (t)/λ. If we The only difference is that in Ac = Aˆ − B ˆ establish that Kc ∈ L∞ , and Ac is u.a.s., then the rest of the proof is identical to that of Theorem 7.4.2. ˆ p Qm , Zˆp guarantees that the soThe strong coprimeness assumption about R > lution P (t) = P (t) > 0 of the Riccati equation exists at each time t and P ∈ L∞ . This, together with the boundedness of the plant parameter estimates, guarantee ˆ and therefore K ˆ c ∈ L∞ . Furthermore, using the results of Section 7.3.4, that B, we can establish that Ac (t) is a stable matrix at each frozen time t. As in Example 7.4.4, we have 2 ˙ ˆ˙ ˆ ˆ 2||B(t)|| ||P (t)|| ||B(t)|| ||P (t)|| ||B(t)|| ˙ˆ ||A˙ c (t)|| ≤ ||A(t)|| + + λ λ

and

˙ P˙ Ac + A> c P = −Q

where

> ˆ˙ B ˆ >P ˆB ˆ˙ P PB ˙ˆ> ˙ˆ P B Q = A P + PA − − λ λ ˙ ˙ which, as shown earlier, imply that kP (t)k and, thus, kAc (t)k ∈ L2 . The pointwise stability of Ac together with kA˙ c (t)k ∈ L2 imply, by Theorem 3.4.11, that Ac is a u.a.s. matrix. The rest of the proof is completed by following exactly the same steps as in the proof of Theorem 7.4.2. 2

7.5

Hybrid APPC Schemes

The stability properties of the APPC schemes presented in Section 7.4 are based on the assumption that the algebraic equations used to calculate the controller parameters are solved continuously and instantaneously. In practice, even with high speed computers and advanced software tools, a short time interval is always required to complete the calculations at a given time t. The robustness and stability properties of the APPC schemes of Section 7.4

7.5. HYBRID APPC SCHEMES

495

Table 7.7 Hybrid adaptive law

yp = Plant

Zp (s) Rp (s) up

Zp (s) = θb∗> αn−1 (s), Rp (s) = sn + θa∗> αn−1 (s) αn−1 (s) = [sn−1 , sn−2 , . . . , s, 1]>

θp(k+1) = θpk + Γ ·

φ= Adaptive law

²=

R tk+1 tk

²(τ )φ(τ )dτ, k = 0, 1, . . .

α> α> n−1 (s) n−1 (s) Λp (s) up , − Λp (s) yp

> φ z−θpk , m2

¸>

,

z=

sn Λp (s) yp

m2 = 1 + φ> φ, ∀t ∈ [tk , tk+1 )

> , θ > ]> θpk = [θbk ak

ˆ p (s, tk ) = sn + θ> R a(k−1) αn−1 (s) > Zˆp (s, tk ) = θb(k−1) αn−1 (s)

Design variables

Ts = tk+1 − tk > Tm ; 2 − Ts λmax (Γ) > γ, for some γ > 0; Λp (s) is monic and Hurwitz with degree n

with respect to such computational real time delays can be considerably improved by using a hybrid adaptive law for parameter estimation. The sampling rate of the hybrid adaptive law may be chosen appropriately to allow for the computations of the control law to be completed within the sampling interval. Let Tm be the maximum time for performing the computations required to calculate the control law. Then the sampling period Ts of the hybrid adaptive law may be chosen as Ts = tk+1 − tk > Tm where {tk : k = 1, 2, . . .} is a time sequence. Table 7.7 presents a hybrid adaptive law based on parametric model (7.4.2). It can be used to replace the continuous-time

496

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL Table 7.8 Hybrid APPC scheme: polynomial approach Zp (s) up Rp (s)

Plant

yp =

Reference signal

Qm (s)ym = 0

Adaptive law

Hybrid adaptive law of Table 7.7

Algebraic equation

ˆ tk ) = sn−1 +l> (tk )αn−2 (s) Solve for L(s, Pˆ (s, tk ) = p> (tk )αn+q−1 (s) from equation ˆ tk )Qm (s)R ˆ p (s, tk ) + Pˆ (s, tk )Zˆp (s, tk ) = A∗ (s) L(s,

Control law

Design variables

up =

ˆ tk )Qm (s) Λ(s) − L(s, Pˆ (s, tk ) up − (yp − ym ) Λ(s) Λ(s)

A∗ monic Hurwitz of degree 2n + q − 1; Qm (s) monic of degree q with nonrepeated roots on the jω axis; Λ(s) = Λp (s)Λq (s); Λp (s), Λq (s) monic and Hurwitz with degree n, q − 1, respectively

adaptive laws of the APPC schemes discussed in Section 7.4 as shown in Tables 7.8 to 7.10. The controller parameters in the hybrid adaptive control schemes of Tables 7.8 to 7.10 are updated at discrete times by solving certain algebraic equations. As in the continuous-time case, the solution of these equations exˆ p (s, tk )Qm (s), Zˆp (s, tk ) are strongly ist provided the estimated polynomials R coprime at each time tk . The following theorem summarizes the stability properties of the hybrid APPC schemes presented in Tables 7.8 to 7.10. ˆ p (s, tk )Qm (s), Zˆp (s, tk ) are Theorem 7.5.1 Assume that the polynomials R strongly coprime at each time t = tk . Then the hybrid APPC schemes given

7.5. HYBRID APPC SCHEMES

497

Table 7.9 Hybrid APPC scheme: state variable approach Plant

yp =

Zp (s) up Rp (s)

Reference signal

Qm (s)ym = 0

Adaptive law

Hybrid adaptive law of Table 7.7.

State observer

ˆk−1 u ˆ eˆ˙ = Aˆk−1 eˆ¯+ B ¯p − K [C > eˆ − e1 ] Aˆk−1 =  o(k−1)  ¯ I ¯ n+q−1  ˆ  ¯ > = −θ1(k−1) ¯ − − −−  B k−1 = θ2(k−1) , C ¯ ¯ 0 ˆ [1, 0, . . . , 0] Ko(k−1) = a∗− θ1(k−1) θ1(k−1) , θ2(k−1) ˆ p (s, tk )Qm (s)−sn+q , are the coefficient vectors of R ˆ Zp (s, tk )Q1 (s), respectively, a∗ is the coefficient vector of A∗o (s) − sn+q

Algebraic equation

ˆ c(k−1) the equation det[sI − Aˆk−1 + Solve for K ˆ c(k−1) ] = A∗c (s) ˆk−1 K B

Control law

ˆ c(k−1) eˆ, up = u ¯p = −K

Design variables

Choose Qm , Q1 , A∗o , A∗c , Q1 as in Table 7.5

Q1 (s) u ¯p Qm (s)

in Tables 7.8 to 7.10 guarantee signal boundedness and convergence of the tracking error to zero asymptotically with time. The proof of Theorem 7.5.1 is similar to that of the theorems in Section 7.4, with minor modifications that take into account the discontinuities in the parameters and is given in Section 7.7. Table 7.10 Hybrid adaptive LQ control scheme

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CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

Zp (s) up Rp (s)

Plant

yp =

Reference signal

Qm (s)ym = 0

Adaptive law

Hybrid adaptive law of Table 7.7.

State observer

Riccati equation

ˆk−1 u ˆ o(k−1) [C > eˆ − e1 ] eˆ˙ = Aˆk−1 eˆ + B ¯p − K ˆ o(k−1) , Aˆk−1 , B ˆk−1 , C as in Table 7.9 K > > 0 the equation Solve for Pk−1 = Pk−1 > ˆk−1B ˆ > Pk−1+CC >= 0 Aˆk−1Pk−1+Pk−1Aˆk−1−λ1 Pk−1B k−1

Control law

Q1 (s) 1 ˆ> ˆ, up = u ¯p u ¯p = − B k−1 Pk−1 e λ Qm (s)

Design variables

Choose λ, Q1 (s), Qm (s) as in Table 7.6

The major advantage of the hybrid adaptive control schemes described in Tables 7.7 to 7.10 over their continuous counterparts is the smaller computational effort required during implementation. Another possible advantage is better robustness properties in the presence of measurement noise, since the hybrid scheme does not respond instantaneously to changes in the system, which may be caused by measurement noise.

7.6

Stabilizability Issues and Modified APPC

The main drawbacks of the APPC schemes of Sections 7.4 and 7.5 is that the adaptive law cannot guarantee that the estimated plant parameters or polynomials satisfy the appropriate controllability or stabilizability condition at each time t, which is required to calculate the controller parameter vector θc . Loss of stabilizability or controllability may lead to computational problems and instability.

7.6. STABILIZABILITY ISSUES AND MODIFIED APPC

499

In this section we concentrate on this problem of the APPC schemes and propose ways to avoid it. We call the estimated plant parameter vector θp at time t stabilizable if the corresponding algebraic equation is solvable for the controller parameters. Because we are dealing with time-varying estimates, uniformity with respect to time is guaranteed by requiring the level of stabilizability to be greater than some constant ε∗ > 0. For example, the level of stabilizability can be defined as the absolute value of the determinant of the Sylvester matrix of the estimated plant polynomials. We start with a simple example that demonstrates the loss of stabilizability that leads to instability.

7.6.1

Loss of Stabilizability: A Simple Example

Let us consider the first order plant y˙ = y + bu

(7.6.1)

where b 6= 0 is an unknown constant. The control objective is to choose u such that y, u ∈ L∞ , and y(t) → 0 as t → ∞. If b were known then the control law 2 u=− y b

(7.6.2)

would meet the control objective exactly. When b is unknown, a natural approach to follow is to use the certainty equivalence control (CEC) law 2 uc = − y ˆb

(7.6.3)

where ˆb(t) is the estimate of b at time t, generated on-line by an appropriate adaptive law. Let us consider the following two adaptive laws: (i) Gradient ˆb˙ = γφε , ˆb(0) = ˆb0 6= 0 where γ > 0 is the constant adaptive gain.

(7.6.4)

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CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

(ii) Pure Least-Squares ˆb˙ = P φε , ˆb(0) = ˆb0 6= 0 φ2 , P (0) = p0 > 0 P˙ = −P 2 1 + β0 φ2

(7.6.5)

where P, φ ∈ R1 , z − ˆbφ , 1 + β0 φ2 z = y˙ f − yf

yf =

ε =

1 1 y, φ= u s+1 s+1 (7.6.6)

It can be shown that for β0 > 0, the control law (7.6.3) with ˆb generated by (7.6.4) or (7.6.5) meets the control objective provided that ˆb(t) 6= 0 ∀t ≥ 0. Let us now examine whether (7.6.4) or (7.6.5) can satisfy the condition ˆb(t) 6= 0, ∀t ≥ 0. From (7.6.1) and (7.6.6), we obtain ε=−

˜bφ 1 + β0 φ2

(7.6.7)

4 where ˜b = ˆb − b is the parameter error. Using (7.6.7) in (7.6.4), we have

ˆb˙ = −γ

φ2 (ˆb − b) , ˆb(0) = ˆb0 1 + β0 φ2

(7.6.8)

Similarly, (7.6.5) can be rewritten as φ2 (ˆb − b) , ˆb(0) = ˆb0 1 + β0 φ2 p0 , p0 > 0 P (t) = R t φ2 1 + p0 0 1+β0 φ2 dτ ˆb˙ = −P

(7.6.9)

˙ It is clear from (7.6.8) and (7.6.9) that for ˆb(0) = b, ˆb(t) = 0 and ˆb(t) = b, ∀t ≥ 0; therefore, the control objective can be met exactly with such an initial condition for ˆb. ˙ If φ(t) = 0 over a nonzero finite time interval, we will have ˆb = 0, u = y = 0, which is an equilibrium state (not necessarily stable though) and the control objective is again met.

7.6. STABILIZABILITY ISSUES AND MODIFIED APPC

501

10 9 8

Output y(t)

7 6 5 4 3 2 1 0

-1

-0.5

0

0.5

1

1.5

2

2.5

Estimate ^b(t)

Figure 7.5 Output y(t) versus estimate ˆb(t) for different initial conditions y(0) and ˆb(0) using the CEC uc = −2y/ˆb. For analysis purposes, let us assume that b > 0 (unknown to the designer). For φ 6= 0, both (7.6.8), (7.6.9) imply that ˙ sgn(ˆb) = −sgn(ˆb(t) − b) and, therefore, for b > 0 we have ˙ ˙ ˆb(t) > 0 if ˆb(0) < b and ˆb(t) < 0 if ˆb(0) > b Hence, for ˆb(0) < 0 < b, ˆb(t) is monotonically increasing and crosses zero leading to an unbounded control uc . Figure 7.5 shows the plots of y(t) vs ˆb(t) for different initial conditions ˆb(0), y(0), demonstrating that for ˆb(0) < 0 < b, ˆb(t) crosses zero leading to unbounded closed-loop signals. The value of b = 1 is used for this simulation. The above example demonstrates that the CEC law (7.6.3) with (7.6.4) or (7.6.5) as adaptive laws for generating ˆb is not guaranteed to meet the control objective. If the sign of b and a lower bound for |b| are known, then the adaptive laws (7.6.4), (7.6.5) can be modified using projection to

502

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

constrain ˆb(t) from changing sign. This projection approach works for this simple example but its extension to the higher order case is awkward due to the lack of any procedure for constructing the appropriate convex parameter sets for projecting the estimated parameters.

7.6.2

Modified APPC Schemes

The stabilizability problem has attracted considerable interest in the adaptive control community and several solutions have been proposed. We list the most important ones below with a brief explanation regarding their advantages and drawbacks. (a) Stabilizability is assumed. In this case, no modifications are introduced and stabilizability is assumed to hold for all t ≥ 0. Even though there is no theoretical justification for such an assumption to hold, it has been often argued that in most simulation studies, no stabilizability problems usually arise. The example presented above illustrates that no stabilizability problem would arise if the initial condition of ˆb(0) happens to be in the region ˆb(0) > b. In the higher order case, loss of stabilizability occurs at certain isolated manifolds in the parameter space when visited by the estimated parameters. Therefore, one can easily argue that the loss of stabilizability is not a frequent phenomenon that occurs in the implementation of APPC schemes. (b) Parameter projection methods [73, 109, 111]. In this approach, the adaptive laws used to estimate the plant parameters on-line are modified using the gradient projection method. The parameter estimates are constrained to lie inside a convex subset C0 of the parameter space that is assumed to have the following properties: (i) The unknown plant parameter vector θp∗ ∈ C0 . (ii) Every member θp of C0 has a corresponding level of stabilizability greater than ²∗ for some known constant ²∗ > 0. We have already demonstrated this approach for the scalar plant yp =

b up s+a

7.6. STABILIZABILITY ISSUES AND MODIFIED APPC

503

In this case, the estimated polynomials are s + a ˆ, ˆb which, for the APPC schemes of Section 7.4 to be stable, are required to be strongly coprime. This implies that ˆb should satisfy |ˆb(t)| ≥ b0 for some b0 > 0 for all t ≥ 0. The subset C0 in this case is defined as n

C0 = ˆb ∈ R1 | ˆbsgn(b) ≥ b0

o

where the unknown b is assumed to belong to C0 , i.e., |b| ≥ b0 . As shown in Examples 7.4.1 and 7.4.2, we guaranteed that |ˆb(t)| ≥ b0 by using projection to constrain ˆb(t) to be inside C0 ∀t ≥ 0. This modification requires that b0 and the sgn(b) are known a priori. Let us now consider the general case of Sections 7.4 and 7.5 where the ˆ p (s, t)Qm (s), Zˆp (s, t) are required to be strongly coestimated polynomials R ˆ p (s, t) prime. This condition implies that the Sylvester matrix Se (θp ) of R ˆ Qm (s), Zp (s, t) satisfies | det Se (θp )| ≥ ²∗ ˆ p (s, t) − sn and where θp ∈ R2n is the vector containing the coefficients of R Zˆp (s, t), and ²∗ > 0 is a constant. If ²∗ > 0 is chosen so that | det Se (θp∗ )| ≥ ²∗ > 0 where θp∗ ∈ R2n is the corresponding vector with the coefficients of the unknown polynomials Rp (s), Zp (s), then the subset C0 may be defined as C0 = convex subset of D ∈ R2n that contains θp∗ where

n

o

D = θp ∈ R2n | | det Se (θp )| ≥ ²∗ > 0

Given such a convex set C0 , the stabilizability of the estimated parameters at each time t is ensured by incorporating a projection algorithm in the adaptive law to guarantee that the estimates are in C0 , ∀t ≥ 0. The projection is based on the gradient projection method and does not alter the usual properties of the adaptive law that are used in the stability analysis of the overall scheme. This approach is simple but relies on the rather strong assumption that the set C0 is known. No procedure has been proposed for constructing such a set C0 for a general class of plants. An extension of this approach has been proposed in [146]. It is assumed that a finite number of convex subsets C1 , . . . , Cp are known such that

504

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

(i) θp∗ ∈ ∪pi=1 Ci and the stabilizability degree of the corresponding plant is greater than some known ε∗ > 0. (ii) For every θp ∈ ∪pi=1 Ci the corresponding plant model is stabilizable with a stabilizability degree greater than ε∗ . In this case, p adaptive laws with a projection, one for each subset Ci , are used in parallel. A suitable performance index is used to select the adaptive law at each time t whose parameter estimates are to be used to calculate the controller parameters. The price paid in this case is the use of p parallel adaptive laws with projection instead of one. As in the case of a single convex subset, there is no effective procedure for constructing Ci , i = 1, 2, . . . , p, with properties (i) and (ii) in general. The assumption, however, that θp∗ ∈ ∪pi=1 Ci is weaker. (c) Correction Approach [40]. In this approach, a subset D in the parameter space is known with the following properties: (i) θp∗ ∈ D and the stabilizability degree of the plant is greater than some known constant ε∗ > 0. (ii) For every θp ∈ D, the corresponding plant model is stabilizable with a degree greater than ε∗ . Two least-squares estimators with estimates θˆp , θ¯p of θp∗ are run in parallel. The controller parameters are calculated from θ¯p as long as θ¯p ∈ D. When θ¯p 6∈ D, θ¯p is reinitialized as follows: θ¯p = θˆp + P 1/2 γ where P is the covariance matrix for the least–squares estimator of θp∗ , and γ is a vector chosen so that θ¯p ∈ D. The search for the appropriate γ can be systematic or random. The drawbacks of this approach are (1) added complexity due to the two parallel estimators, and (2) the search procedure for γ can be tedious and time-consuming. The advantages of this approach, when compared with the projection one, is that the subset D does not have to be convex. The

7.6. STABILIZABILITY ISSUES AND MODIFIED APPC

505

importance of this advantage, however, is not clear since no procedure is given as to how to construct D to satisfy conditions (i), (ii) above. (d) Persistent excitation approach [17, 49]. In this approach, the reference input signal or an external signal is chosen to be sufficiently rich in frequencies so that the signal information vector is PE over an interval. The PE property guarantees that the parameter estimate θˆp of θp∗ converges exponentially to θp∗ (provided the covariance matrix in the case of least squares is prevented from becoming singular). Using this PE property, and by assuming that a lower bound ε∗ > 0 for the stabilizability degree of the plant is known, the following modification is used: When the stabilizability degree of the estimated plant is greater than ε∗ , the controller parameters are computed using θˆp ; otherwise the controller parameters are frozen to their previous value. Since θˆp converges to θp∗ , the stabilizability degree of the estimated plant is guaranteed to be greater than ε∗ asymptotically with time. The main drawback of this approach is that the reference signal or external signal has to be on all the time, in addition to being sufficiently rich, which implies that accurate regulation or tracking of signals that are not rich is not possible. Thus the stabilizability problem is overcome at the expense of destroying the desired tracking or regulation properties of the adaptive scheme. Another less serious drawback is that a lower bound ε∗ > 0 for the stabilizability degree of the unknown plant is assumed to be known a priori. An interesting method related to PE is proposed in [114] for the stabilization of unknown plants. In this case the PE property of the signal information vector over an interval is generated by a “rich” nonlinear feedback term that disappears asymptotically with time. The scheme of [114] guarantees exact regulation of the plant output to zero. In contrast to other PE methods [17, 49], both the plant and the controller parameters are estimated on-line leading to a higher order adaptive law. (e) The cyclic switching strategy [166]. In this approach the control input is switched between the CEC law and a finite member of specially constructed controllers with fixed parameters. The fixed controllers have the property that at least one of them makes the resulting closed-loop plant observable through a certain error signal. The switching logic is based on a cyclic switching rule. The proposed scheme does not rely on persistent excitation

506

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

and does not require the knowledge of a lower bound for the level of stabilizability. One can argue, however, that the concept of PE to help cross the points in the parameter space where stabilizability is weak or lost is implicitly used by the scheme because the switching between different controllers which are not necessarily stabilizing may cause considerable excitation over intervals of time. Some of the drawbacks of the cyclic switching approach are the complexity and the possible bad transient of the plant or tracking error response during the initial stages of adaptation when switching is active. (f ) Switched-excitation approach [63]. This approach is based on the use of an open loop rich excitation signal that is switched on whenever the calculation of the CEC law is not possible due to the loss of stabilizability of the estimated plant. It differs from the PE approach described in (d) in that the switching between the rich external input and the CEC law terminates in finite time after which the CEC law is on and no stabilizability issues arise again. We demonstrate this method in subsection 7.6.3. Similar methods as above have been proposed for APPC schemes for discrete-time plants [5, 33, 64, 65, 128, 129, 189, 190].

7.6.3

Switched-Excitation Approach

Let us consider the same plant (7.6.1) and control objective as in subsection 7.6.1. Instead of the control law (7.6.3), let us propose the following switching control law (

u=

−2y/ˆb c

if t ∈ [0, t1 ) ∪ (tk + jk τ, tk+1 ) if t ∈ [tk , tk + jk τ ]

(7.6.10)

where c 6= 0 is a constant. ˆb˙ = P φε, ˆb(0) = ˆb0 6= 0

(7.6.11)

P 2 φ2 P˙ = − , P (0) = P (tk ) = P (tk + jτ ) = p0 I > 0, j = 1, 2, . . . , jk 1 + β0 φ2 (

β0 =

1 0

if t ∈ [0, t1 ) ∪ (tk + jk τ, tk+1 ) if t ∈ [tk , tk + jk τ ]

where ε, φ are as defined in (7.6.6), and

7.6. STABILIZABILITY ISSUES AND MODIFIED APPC

507

• k = 1, 2, . . . • t1 is the first time instant for which |ˆb(t1 )| = ν(1) where ν(k) =

|ˆb0 | ν(1) = , k = 1, 2, . . . 2k k

• tk is the first time instant after t = tk−1 + jk−1 τ for which |ˆb(tk )| = ν(k +

k X i=1

ji ) =

ν(1) P k + ki=1 ji

where jk = 1, 2, . . . is the smallest integer for which |ˆb(tk + jk τ )| > ν(k +

k X

ji )

i=1

and τ > 0 is a design constant. Even though the description of the above scheme appears complicated, the intuition behind it is very simple. We start with an initial guess ˆb0 6= 0 for ˆb and apply u = uc = −2y/ˆb. If |ˆb(t)| reaches the chosen threshold ν(1) = |ˆb0 |/2, say at t = t1 , u switches from u = uc to u = ur = c 6= 0, where ur is a rich signal for the plant considered. The signal ur is applied for an interval τ , where τ is a design constant, and |ˆb(t1 + τ )| is compared with the new threshold ν(2) = ν(1)/2. If |ˆb(t1 +τ )| > ν(2), then we switch back to u = uc at t = t1 + τ. We continue with u = uc unless |ˆb(t2 )| = ν(3) = ν(1)/3 for some finite t2 in which case we switch back to u = ur . If |ˆb(t1 +τ )| ≤ ν(2), we continue with u = ur until t = t1 + 2τ and check for the condition ν(1) |ˆb(t1 + 2τ )| > ν(3) = 3

(7.6.12)

If ˆb(t1 + 2τ ) satisfies (7.6.12), we switch to u = uc and repeat the same procedure by reducing the threshold ν(k) at each step. If ˆb(t1 + 2τ ) does not satisfy (7.6.12) then we continue with u = ur for another interval τ and check for ν(1) |ˆb(t1 + 3τ )| > ν(4) = 4

508

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

and repeat the same procedure. We show that the sequences tk , ν(k) converge in a finite number of steps, and therefore |ˆb(t)| > ν ∗ > 0 for some constant ν ∗ and u = uc for all t greater than some finite time t∗ . In the sequel, for the sake of simplicity and without loss of generality, we take c = 1, p0 τ = 1 and adjust the initial condition φ(tk ) for the filter 1 φ = s+1 u to be equal to φ(tk ) = 1 so that φ(t) = 1 for t ∈ [tk , tk + jk τ ] and ∀k ≥ 1. Let us start with a “wrong” initial guess for ˆb(0), i.e., assume that b > 0 and take ˆb(0) < 0. Because ˆb˙ = −P φ2 (ˆb − b)/(1 + β0 φ2 ) ˙ and P (t) > 0, for any finite time t, we have that ˆb ≥ 0 for ˆb(t) < b where ˙ ˆb(t) = 0 if and only if φ(t) = 0. Because φ(t) = 1 ∀t ≥ 0, ˆb(t), starting from ˆb(0) < 0, is monotonically increasing. As ˆb(t) increases, approaching zero, it satisfies, at some time t = t1 , ˆb(t1 ) = −ν(1) = −|ˆb0 |/2 and, therefore, signals the switching of the control law from u = uc to u = ur , i.e., for t ≥ t1 , we have u = ur = 1 P˙ = −P 2 φ2 , P (t1 ) = p0

(7.6.13)

ˆb˙ = −P φ2 (ˆb − b)

(7.6.14)

The solutions of (7.6.13) and (7.6.14) are given by P (t) = ˆb(t) = b +

p0 , t ≥ t1 1 + p0 (t − t1 )

1 (ˆb(t1 ) − b), t ≥ t1 1 + p0 (t − t1 )

We now need to monitor ˆb(t) at t = t1 + j1 τ, j1 = 1, 2, . . . until ¯

¯

¯ ν(1) + b ¯¯ ν(1) |ˆb(t1 + j1 τ )| = ¯¯b − (because p0 τ = 1) > ¯ 1 + j1 1 + j1

(7.6.15)

7.6. STABILIZABILITY ISSUES AND MODIFIED APPC

509

is satisfied for some j1 and switch to u = uc at t = t1 + j1 τ . We have bj1 − ν(1) |ˆb(t1 + j1 τ )| = 1 + j1

(7.6.16)

Let j1∗ be the smallest integer for which bj1∗ > ν(1). Then, ∀j1 ≥ j1∗ , condition (7.6.15) is the same as bj1 − ν(1) ν(1) |ˆb(t1 + j1 τ )| = ˆb(t1 + j1 τ ) = > 1 + j1 1 + j1

(7.6.17)

Hence, for j1 = 2j1∗ , (7.6.17) is satisfied, i.e., by applying the rich signal u = ur = 1 for 2j1∗ intervals of length τ , ˆb(t) passes through zero and ˙ exceeds the value of ν ∗ = ν(1)/(1 + 2j1∗ ) > 0. Because ˆb(t) ≥ 0, we have ˆb(t) > ν ∗ , ∀t ≥ t1 + 2j ∗ τ and therefore u = uc = −2y/ˆb, without any further 1

switching. Figure 7.6 illustrates typical time responses of y, u and ˆb when the switched–excitation approach is applied to the first order plant given by (7.6.1). The simulations are performed with b = 1 and ˆb(0) = −1.5 . At t = t1 ≈ 0.2s, ˆb(t1 ) = ν(1) = ˆb(0)/2 = −0.75, the input u = uc = −2y/ˆb is switched to u = ur = 1 for a period τ = 0.25s. Because at time t = t1 + τ, ˆb(t1 + τ ) is less than ν(2) = ν(1)/2, u = ur = 1 is applied for another period τ. Finally, at time t = t1 + 2τ , ˆb(t) > ν(3) = ν(1)/3, therefore u is switched back to u = uc . Because ˆb(t) > ν(1)/3, ∀t ≥ 0.7s, no further switching occurs and the exact regulation of y to zero is achieved.

General Case The above example may be extended to the general plant (7.3.1) and control objectives defined in the previous subsections as follows. Let uc denote the certainly equivalence control law based on any one of the approaches presented in Section 7.4. Let 4

Cd (θp ) = det |Se (θp )|

510

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

Output y(t)

4 3 2 1 0

0

0.5

1

1.5

2 Time (sec)

2.5

3

3.5

4

Control input u(t)

5 0 -5 -10

uc

ur

ur uc

-15 -20

0

0.2

0.4

0.6

0.8

1 1.2 Time (sec)

1.4

1.6

1.8

2

1.4

1.6

1.8

2

1

Estimate ^b(t)

0.5

0

-0.5

-1

-1.5

0

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

Figure 7.6 The time response of the output y(t), control input u(t) and estimate ˆb(t) for the switched-excitation approach. where Se (θp ) denotes the Sylvester matrix of the estimated polynomials ˆ p (s, t) Qm (s), Zˆp (s, t) and θp (t) is the vector with the plant parameter R estimates. Following the scalar example, we propose the modified control law (

u=

uc (t) ur (t)

if t ∈ [0, t1 ) ∪ (tk + jk τ, tk+1 ) if t ∈ [tk , tk + jk τ ]

(7.6.18)

7.6. STABILIZABILITY ISSUES AND MODIFIED APPC

511

Adaptive Law θ˙p = P εφ,

θp (0) = θ0

(7.6.19)

P φφ> P , P (0) = P0 = P0> > 0 P˙ = − 1 + β0 φ> φ where ε = (z − θp> φ)/(1 + β0 φ> φ) and θ0 is chosen so that Cd (θ0 ) > 0. Furthermore, P (0) = P (tk ) = P (tk + jτ ) = k0−1 I, j = 1, 2, . . . , jk

(7.6.20)

where k0 = constant > 0, and (

β0 =

1 0

if t ∈ [0, t1 ) ∪ (tk + jk τ, tk+1 ) if t ∈ [tk , tk + jk τ ]

(7.6.21)

where • k = 1, 2, . . . • t1 is the first time instant for which Cd (θp (t1 )) = ν(1) > 0, where ν(k) =

ν(1) Cd (θ0 ) = , k = 1, 2, . . . 2k k

(7.6.22)

• tk (k ≥ 2) is the first time instant after t = tk−1 + jk−1 τ for which Cd (θp (tk )) = ν(k +

k−1 X i=1

ji ) =

ν(1) (k +

Pk−1 i=1

ji )

(7.6.23)

and jk = 1, 2, . . . is the smallest integer for which Cd (θp (tk + jk τ )) > ν(k +

k X

ji )

(7.6.24)

i=1

where τ > 0 is a design constant. • uc (t) is the certainty equivalence control given in Section 7.4. • ur (t) is any bounded stationary signal which is sufficiently rich of order 2n. For example, one can choose ur (t) =

n X i=1

Ai sinωi t

512

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

where Ai 6= 0, i = 1, . . . , n, and ωi 6= ωj for i 6= j. From (7.6.18), we see that in the time intervals (tk + jk τ, tk+1 ), the staP bilizability degree Cd (θp (t)) is above the threshold ν(k + ki=1 ji ) and the adaptive control system includes a normalized least-squares estimator and a pole placement controller. In the time intervals [tk+1 , tk+1 + jk+1 τ ], the control input is equal to an external exciting input ur (t) and the parameter vector estimate θp (t) is generated by an unnormalized least-squares estimator. The switching (at time t = tk ) from the pole placement control uc to the external rich signal ur occurs when the stabilizability degree Cd (θp (t)) of the P estimated model reaches the threshold ν(k + ki=1 ji ). We keep applying u = ur during successive time intervals of fixed length τ , until time t = tk + jk τ P for which the condition Cd (θp (tk + jk τ )) > ν(k + ki=1 ji ) is satisfied and u switches back to the CEC law. The idea behind this approach is that when the estimated model is stabilizable, the control objective is pole placement and closed-loop stabilization; but when the estimation starts to deteriorate, the control priority becomes the “improvement” of the quality of estimation, so that the estimated parameters can cross the hypersurfaces that contain the points where Cd (θp ) is close to zero. The following theorem establishes the stability properties of the proposed adaptive pole placement scheme. Theorem 7.6.1 All the signals in the closed-loop (7.3.1) and (7.6.18) to (7.6.24) are bounded and the tracking error converges to zero as t → ∞. Furthermore, there exist finite constants ν ∗ , T ∗ > 0 such that for t ≥ T ∗ , we have Cd (θp (t)) ≥ ν ∗ and u(t) = uc . The proof of Theorem 7.6.1 is rather long and can be found in [63]. The design of the switching logic in the above modified controllers is based on a simple and intuitive idea that when the quality of parameter estimation is “poor,” the objective changes from pole placement to parameter identification. Parameter identification is aided by an external open-loop sufficiently rich signal that is kept on until the quality of the parameter estimates is acceptable for control design purposes. One of the advantages of the switched-excitation algorithm is that it is intuitive and easy to implement. It may suffer, however, from the same drawbacks as other switching algorithms, that is, the transient performance may be poor during switching.

7.7. STABILITY PROOFS

513

The adaptive control scheme (7.6.18) to (7.6.24) may be simplified when a lower bound ν ∗ > 0 for Cd (θp∗ ) is known. In this case, ν(k) = ν ∗ ∀k. In the proposed scheme, the sequence ν(k) converges to ν ∗ and therefore the lower bound for Cd (θp∗ ) is also identified. The idea behind the identification of ν ∗ is due to [189] where very similar to the switched-excitation approach methods are used to solve the stabilizability problem of APPC for discrete-time plants.

7.7

Stability Proofs

In this section we present all the long proofs of the theorems of the previous subsections. In most cases, these proofs follow directly from those already presented for the simple examples and are repeated for the sake of completeness.

7.7.1

Proof of Theorem 7.4.1

Step 1. Let us start by establishing the expressions (7.4.24). We rewrite the control law (7.4.23) and the normalized estimation error as ˆ m 1 up = −Pˆ 1 (yp − ym ) LQ Λ Λ ˆp ²m2 = z − θp> φ = R

1 1 yp − Zˆp up Λp Λp

(7.7.1) (7.7.2)

where Λ(s), Λp (s) are monic, Hurwitz polynomials of degree n+q−1, n, respectively, ˆ p = sn +θa> αn−1 (s), Zˆp = θ> αn−1 (s). From Table 7.4, we have Λ(s) = Λp (s)Λq (s), R b where Λq (s) is a monic Hurwitz polynomial of degree q−1. This choice of Λ simplifies the proof. We should point out that the same analysis can also be carried out with Λ, Λp being Hurwitz but otherwise arbitrary, at the expense of some additional algebra. Let us define 4 1 4 1 uf = up , yf = yp Λ Λ and write (7.7.1), (7.7.2) as ˆ m uf = ym1 , R ˆ p Λq yf − Zˆp Λq uf = ²m2 Pˆ yf + LQ

(7.7.3)

4 ˆ p (s)Λq (s), Zˆp (s)Λq (s), where ym1 = Pˆ Λ1 ym ∈ L∞ . By expressing the polynomials R ˆ Pˆ (s), L(s)Q m (s) as

ˆ p (s)Λq (s) = sn+q−1 + θ¯1> αn+q−2 (s), Zˆp (s)Λq (s) = θ¯2> αn+q−2 (s) R

514

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL n+q−1 ˆ Pˆ (s) = p0 sn+q−1 + p¯> αn+q−2 (s), L(s)Q + ¯l> αn+q−2 (s) m (s) = s

we can rewrite (7.7.3) in the form of (n+q−1)

= −θ¯1> αn+q−2 (s)yf + θ¯2> αn+q−2 (s)uf + ²m2

(n+q−1)

= (p0 θ¯1 − p¯)> αn+q−2 (s)yf − (p0 θ¯2 + ¯l)> αn+q−2 (s)uf

yf

uf

(7.7.4)

2

−p0 ²m + ym1 (n+q−1)

where the second equation is obtained from (7.7.3) by substituting for yf h i> 4 (n+q−2) (n+q−2) Defining the state x = yf , . . . , y˙ f , yf , uf , . . . , u˙ f , uf , we obtain x˙ = A(t)x + b1 (t)²m2 + b2 ym1 where



−θ¯1> −−−−− ¯ −− ¯ 0 ¯ ¯ In+q−2 ¯ ... ¯ ¯ 0 −−−−−−− p0 θ¯1> − p¯> −−−−−−−

         A(t)=         O(n+q−2)×(n+q−1)

|

|

|

|

.

(7.7.5)

 θ¯2> −−−−−−−        1 0    ..   ..  O(n+q−2)×(n+q−1)   .  .        0  0          − − − − − − −  , b1 (t) =  −p0  , b2 =  1 > > ¯ ¯      −p0 θ2 − l   0  0     . . −−−−−  ..   ..  ¯ −−   ¯ 0  ¯ 0 0  ¯ In+q−2 ¯ ...  ¯ ¯ 0

O(n+q−2)×(n+q−1) is an (n + q − 2) by (n + q − 1) matrix with all elements equal (n+q−1) to zero. Now, because up = Λuf = uf + λ> αn+q−2 (s)uf , yp = Λyf = (n+q−1)

yf have

+ λ> αn+q−2 (s)yf where λ is the coefficient vector of Λ(s) − sn+q−1 , we up

=

[0, . . . , 0, 1, 0, . . . , 0]x˙ + [0, . . . , 0, λ> ]x | {z } | {z } | {z } n+q−1

yp

=

n+q−1

n+q−1

[1, 0, . . . , 0, 0, . . . , 0]x˙ + [λ> , 0, . . . , 0]x | {z } | {z } | {z } n+q−1

n+q−1

(7.7.6)

n+q−1

ˆ Step 2. Establish the e.s. of the homogeneous part of (7.7.5). Because Pˆ , L 1 satisfy the Diophantine equation (7.4.21), we can show that for each frozen time t, ˆ p Λq · LQ ˆ m + Pˆ · Zˆp Λq = A∗ Λq det(sI − A(t)) = R 1

(7.7.7)

X · Y denotes the algebraic product of two polynomials that may have time-varying coefficients.

7.7. STABILITY PROOFS

515

i.e., A(t) is a stable matrix for each frozen time t. One way to verify (7.7.7) is to ˆ p , Zˆp , Pˆ , L ˆ frozen at each time t. It follows consider (7.7.3) with the coefficients of R from (7.7.3) that yf =

1 ˆ m ²m2 + Zˆp Λq ym1 ) (LQ ˆ ˆ Rp Λq · LQm + Pˆ · Zˆp Λq

whose state space realization is given by (7.7.5). Because ˆ p Λq · LQ ˆ m + Pˆ · Zˆp Λq = A∗ Λq R (7.7.7) follows. ˙ We now need to show that kA(t)k ∈ L2 , kA(t)k ∈ L∞ from the properties θp ∈ L∞ and θ˙p ∈ L2 which are guaranteed by the adaptive law of Table 7.4. Using ˆ p Qm and Zˆp are strongly coprime at each the assumption that the polynomials R time t, we conclude that the Sylvester matrix Sˆl (defined in Table 7.4) is uniformly nonsingular, i.e., | det(Sˆl )| > ν0 for some ν0 > 0, and thus θp ∈ L∞ implies that Sl , Sl−1 ∈ L∞ . Therefore, the solution βˆl of the algebraic equation Sˆl βˆl = αl∗ which can be expressed as βˆl = Sˆl−1 αl∗ is u.b. On the other hand, because θp ∈ L∞ and θ˙p ∈ L2 , it follows from the ˙ definition of the Sylvester matrix that kSˆl (t)k ∈ L2 . Noting that ˙ ˙ βˆl = −Sˆl−1 Sˆl Sˆl−1 αl∗ ˙ ˙ we have βˆl ∈ L2 which is implied by Sˆl , Sˆl−1 ∈ L∞ and kSˆl (t)k ∈ L2 . Because the vectors θ¯1 , θ¯2 , p¯, ¯l are linear combinations of θp , βˆl and all elements in A(t) are uniformly bounded, we have ˙ ˙ kA(t)k ≤ c(|βˆl (t)| + |θ˙p (t)|) ˙ which implies that kA(t)k ∈ L2 . Using Theorem 3.4.11, it follows that the homogeneous part of (7.7.5) is e.s. Step 3. Use the properties of the L2δ norm and B-G Lemma to establish boundedness. As before, for clarity of presentation, we denote the L2δ norm as k · k, then from Lemma 3.3.3 and (7.7.5) we have kxk ≤ ck²m2 k + c, |x(t)| ≤ ck²m2 k + c

(7.7.8)

4

for some δ > 0. Defining m2f = 1 + kyp k2 + kup k2 , it follows from (7.7.5), (7.7.6), (7.7.8) and Lemma 3.3.2 that φ, m ≤ mf and m2f ≤ c + ckxk2 + ck²m2 k2 ≤ ck²mmf k2 + c

(7.7.9)

516

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

i.e.,

Z m2f ≤ c + c

0

t

e−δ(t−τ ) ²2 (τ )m2 (τ )m2f (τ )dτ

Because ²m ∈ L2 , guaranteed by the adaptive law, the boundedness of mf can be established by applying the B-G Lemma. The boundedness of the rest of the signals follows from mf ∈ L∞ and the properties of the L2δ norm that is used to show that mf bounds most of the signals from above. Step 4. Establish that the tracking error converges to zero. The tracking properties of the APPC scheme are established by manipulating (7.7.1) and (7.7.2) as follows: From the normalized estimation error equation we have ˆp ²m2 = R

1 1 yp − Zˆp up Λp Λp

4 n−1 > ¯ m 1 where L(s, ¯ t) = Filtering each side of the above equation with LQ s +αn−2 (s)l Λq > > ˆ and lc = [1, l ] is the coefficient vector of L(s, t), it follows that

¯ m 1 ¯ m 1 (²m2 ) = LQ LQ Λq Λq

µ

ˆ p 1 yp − Zˆp 1 up R Λp Λp

¶ (7.7.10)

Noting that Λ = Λq Λp , and applying the Swapping Lemma A.1, we obtain the following equations: µ ¶ Qm ˆ 1 ˆ p Qm yp + r1 , Qm Zˆp 1 up = Zˆp Qm up + r2 Rp yp = R (7.7.11) Λq Λp Λ Λq Λp Λ where µµ 4

r1 = Wc1 (s)

¶ ¶ ¶ ¶ µµ > > (s) αn−1 (s) αn−1 4 ˙ Wb1 (s) yp θa , r2 = Wc1 (s) up θ˙b Wb1 (s) Λp Λp

and Wc1 , Wb1 are as defined in Swapping Lemma A.1 with W = QΛm . Because q up , yp ∈ L∞ and θ˙a , θ˙b ∈ L2 , it follows that r1 , r2 ∈ L2 . Using (7.7.11) in (7.7.10), we have µ ¶ 1 Qm Qm 2 ˆ ¯ ¯ ˆ LQm (²m ) = L Rp yp − Zp up + r1 − r2 (7.7.12) Λq Λ Λ Noting that Qm yp = Qm (e1 + ym ) = Qm e1 , we can write (7.7.12) as µ ¶ µ ¶ 1 Qm Qm 2 ¯ ¯ ˆ ¯ ˆ LQm (²m ) = L Rp e1 + r1 − r2 − L Zp up Λq Λ Λ

(7.7.13)

7.7. STABILITY PROOFS

517

Applying the Swapping Lemma A.4 (i) to the second term in the right-hand side of (7.7.13), we have µ ¶ µ ¶ Qm Qm ¯ ¯ ˆ ˆ up = Zp L(s, t) u p + r3 (7.7.14) L Zp Λ Λ (by taking

Qm Λ up

> = f in Lemma A.4 (i)), where Z¯p = αn−1 (s)θb and > F (l, θb )αn−2 (s) r3 = αn−2

Qm up Λ(s)

where kF (l, θb )k ≤ c1 |l|˙ + c2 |θ˙b | for some constants c1 , c2 > 0. Because ˆ t)Qm (s) = L(s, ˆ t) · Qm (s) L(s, we use the control law (7.7.1) to write ˆ t) · Qm (s) up = −Pˆ 1 e1 ˆ t) Qm (s) up = L(s, L(s, Λ(s) Λ(s) Λ(s) Substituting (7.7.15) in (7.7.14) and then in (7.7.13), we obtain µ ¶ µ ¶ ˆ p Qm e1 + Z¯p Pˆ 1 e1 + L(r ¯ m 1 (²m2 ) = L ¯ R ¯ 1 − r2 ) − r3 LQ Λq Λ(s) Λ(s)

(7.7.15)

(7.7.16)

According to Swapping Lemma A.4 (ii) (with f = Λ1 e1 and Λ0 (s) = 1), we can write µ ¶ ˆ p Qm e1 ¯ R ˆ t) · R ˆ p (s, t)Qm (s) 1 e1 + r4 L = L(s, Λ(s) Λ(s) ¶ µ 1 1 e1 = Zˆp (s, t) · Pˆ (s, t) e1 + r5 (7.7.17) Z¯p Pˆ Λ(s) Λ(s) where 4

4

0

> > r4 = αn−1 (s)G(s, e1 , l, θa ), r5 = αn−1 (s)G (s, e1 , θb , p) 0

(7.7.18)

and G(s, e1 , l, θa ), G (s, e1 , θb , p) are defined in the Swapping Lemma A.4 (ii). Be0 cause e1 ∈ L∞ and l,˙ p, ˙ θ˙a , θ˙b ∈ L2 , it follows from the definition of G, G that 0 G, G ∈ L2 . Using (7.7.17) in (7.7.16) we have ³ ´ 1 ¯ m 1 (²m2 ) = L(s, ˆ t) · R ˆ p (s, t)Qm (s)+ Zˆp · Pˆ (s, t) ¯ 1−r2 )−r3+r4+r5 LQ e1 + L(r Λq Λ(s) (7.7.19) ¯ t) to denote the swapped polynomial of X(s, t), In the above equations, we use X(s, i.e., 4 4 > ¯ X(s, t) = p> x (t)αn (s), X(s, t) = αn (s)px (t)

518

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

Because we have

ˆ t) · R ˆ p (s, t)Qm (s) + Zˆp (s, t) · Pˆ (s, t) = A∗ (s) L(s, ˆ t) · R ˆ p (s, t)Qm (s) + Zˆp (s, t) · Pˆ (s, t) = A∗ (s) = A∗ (s) L(s,

where the second equality holds because the coefficients of A∗ (s) are constant. Therefore, (7.7.19) can be written as A∗ (s)

1 e1 = v Λ(s)

i.e., e1 = where

µ ¯ v = L Qm 4

Λ(s) v A∗ (s)

1 (²m2 ) − r1 + r2 Λq (s)

(7.7.20) ¶ + r3 − r4 − r5

> ¯ (s)lc , lc = [1, l> ]> we have Because L(s) = αn−1 > v = αn−1 (s)lc [

Qm (²m2 ) − r1 + r2 ] + r3 − r4 − r5 Λq

Therefore, it follows from (7.7.20) that e1 =

Λ(s) > Qm (s) α (s)lc (²m2 ) + v0 A∗ (s) n−1 Λq (s)

(7.7.21)

where v0

=

v1

=

v2

=

Λ(s)α>

Λ(s) > > > [α (s)lc (r2 − r1 ) + αn−2 (s)v1 − αn−1 (s)v2 ] A∗ (s) n−1 Qm (s) F (l, θb )αn−2 (s) up Λ(s) 0

G(s, e1 , l, θa ) + G (s, e1 , p, θb )

(s)

(s)Qm (s) n−1 Because , αn−2Λ(s) are strictly proper and stable, lc ∈ L∞ , and A∗ (s) r1 , r2 , v1 , v2 ∈ L2 , it follows from Corollary 3.3.1 that v0 ∈ L2 and v0 (t) → 0 as t → ∞. Applying the Swapping Lemma A.1 to the first term on the right side of (7.7.21) we have

e1 = lc>

αn−1 (s)Λ(s)Qm (s) (²m2 ) + Wc (s)(Wb (s)(²m2 ))l˙c + v0 A∗ (s)Λq (s)

7.7. STABILITY PROOFS

519

Qm Λ where Wc (s), Wb (s) have strictly proper stable elements. Because αn−1 is A ∗ Λq T T T proper, l˙c , ²m ∈ L∞ L2 and v0 ∈ L∞ L2 , it follows that e1 ∈ L∞ L2 . Z The plant equation yp = Rpp up assumes a minimal state space representation of the form Y˙ = AY + Bup

yp = C > Y where (C, A) is observable due to the coprimeness of Zp , Rp . Using output injection we have Y˙ = (A − KC > )Y + Bup + Kyp 4

where K is chosen so that Aco = A − KC > is a stable matrix. Because yp , up ∈ L∞ and Aco is stable, we have Y ∈ L∞ , which implies that Y˙ , y˙ p ∈ L∞ . Therefore, e˙ 1 = y˙ p − y˙ m ∈ L∞ , which, together with e1 ∈ L2 and Lemma 3.2.5, implies that e1 (t) → 0 as t → ∞.

7.7.2

Proof of Theorem 7.4.2

Step 1. Develop the closed-loop state error equations. We start by representing the tracking error equation Zp Q1 e1 = u ¯p Rp Qm in the following state-space form ¯   ¯ In+q−1 ¯ ∗ e˙ = −θ1 ¯¯ − − −−  e + θ2∗ u ¯p , e ∈ Rn+q ¯ 0

(7.7.22)

e1 = C > e where C > = [1, 0, . . . , 0] ∈ Rn+q and θ1∗ , θ2∗ are the coefficient vectors of Rp Qm − sn+q , Zp Q1 , respectively. 4

Let eo = e − eˆ be the state observation error. Then from the equation for eˆ in Table 7.5 and (7.7.22), we have eˆ˙ = e˙ o = where

ˆ o C > eo Ac (t)ˆ e+K Ao eo + θ˜1 e1 − θ˜2 u ¯p

¯  ¯ In+q−1 ¯ 4 Ao = −a∗ ¯¯ − − −−  ¯ 0 

(7.7.23)

520

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

4 4 4 ˆK ˆ c and θ˜1 = is a stable matrix; Ac (t) = Aˆ − B θ1 − θ1∗ , θ˜2 = θ2 − θ2∗ . The plant input and output satisfy

yp

=

up

=

C > eo + C > eˆ + ym Rp XQ1 Q1 Y Rp u ¯p + yp A∗ A∗

(7.7.24)

where X(s), Y (s) are polynomials of degree n + q − 1 that satisfy (7.3.25) and A∗ (s) is a Hurwitz polynomial of degree 2(n+q)−1. Equation (7.7.24) for up is established in the proof of Theorem 7.3.1 and is used here without proof. Step 2. Establish e.s. for the homogeneous part of (7.7.23). Let us first examine the stability properties of Ac (t) in (7.7.23). For each frozen time t, we have ˆK ˆ c ) = A∗c (s) det(sI − Ac ) = det(sI − Aˆ + B (7.7.25) ˆ p (s, t)Qm (s) are strongly i.e., Ac (t) is stable at each frozen time t. If Zˆp (s, t)Q1 (s), R ˆ B) ˆ is strongly controllable at each time t, then the controller gains coprime, i.e., (A, ˆ Kc may be calculated at each time t using Ackermann’s formula [95], i.e., ˆ c = [0, 0, . . . , 0, 1]Gc−1 A∗c (A) ˆ K where

4

ˆ AˆB, ˆ . . . , Aˆn+q−1 B] ˆ Gc = [B, ˆ B). ˆ Because (A, ˆ B) ˆ is assumed to be is the controllability matrix of the pair (A, ˆ B ˆ ∈ L∞ due to θp ∈ L∞ , we have K ˆ c ∈ L∞ . Now, strongly controllable and A, ½ ¾ ˙ −1 ∗ ˆ −1 d −1 ˙ ∗ ˆ ˆ K c = [0, 0, . . . , 0, 1] −Gc Gc Gc Ac (A) + Gc A (A) dt c ˆ˙ c (t)k ∈ L2 , which, in turn, implies Because θp ∈ L∞ and θ˙p ∈ L2 , it follows that kK ˙ that kAc (t)k ∈ L2 . From Ac being pointwise stable and kA˙ c (t)k ∈ L2 , we have that ˆ p (s, t)Qm (s) Ac (t) is a u.a.s matrix by applying Theorem 3.4.11. If Zˆp (s, t)Q1 (s), R ˆ p (s, t)Qm (s) are, the boundedness of K ˆc are not strongly coprime but Zˆp (s, t), R ˙ˆ ˆ B) ˆ into the strongly and kK c (t)k ∈ L2 can still be established by decomposing (A, controllable and the stable uncontrollable or weakly controllable parts and using ˆ c . Because Ao is a stable matrix the the results in [95] to obtain an expression for K homogeneous part of (7.7.23) is e.s. Step 3. Use the properties of the L2δ norm and the B-G Lemma to establish boundedness. As in Example 7.4.2., we apply Lemmas 3.3.3, 3.3.2 to (7.7.23) and (7.7.24), respectively, to obtain kˆ ek ≤ kyp k ≤ kup k ≤

ckC > eo k ckC > eo k + ckˆ ek + c ≤ ckC > eo k + c >

ckˆ ek + ckyp k ≤ ckC eo k + c

(7.7.26)

7.7. STABILITY PROOFS

521

where k · k denotes the L2δ norm for some δ > 0. We relate the term C > eo with the estimation error by using (7.7.23) to express > C eo as C > eo = C > (sI − Ao )−1 (θ˜1 e1 − θ˜2 u ¯p ) (7.7.27) Noting that (C, Ao ) is in the observer canonical form, i.e., C > (sI − Ao )−1 = αn+q−1 (s) A∗ (s) , we have o

>

C eo =

n ¯ X sn¯ −i i=0

A∗o

(θ˜1i e1 − θ˜2i u ¯p ),

n ¯ =n+q−1

where θ˜i = [θ˜i1 , θ˜i2 , . . . , θ˜i¯n ]> , i = 1, 2. Applying Swapping Lemma A.1 to each term under the summation, we have µ ¶ sn¯ −i ˜ Λp (s)Q1 (s) ˜ sn¯ −i ˙ ˜ θ1i e1 = θ1i e1 + Wci (s) (Wbi (s)e1 ) θ1i A∗o (s) A∗o (s) Λp (s)Q1 (s) and Λp (s)Q1 (s) sn¯ −i ˜ θ2i u ¯p = ∗ Ao (s) A∗o (s)

µ θ˜2i

sn¯ −i ˙ u ¯p + Wci (s) (Wbi (s)¯ up ) θ˜2i Λp (s)Q1 (s)

¶

where Wci , Wbi , i = 0, . . . , n + q − 1 are transfer matrices defined in Lemma A.1 ¯ sn−i with W (s) = Λp (s)Q . Therefore, C > eo can be expressed as 1 (s) C > eo

= =

where

¶ n ¯ µ sn¯ −i sn¯ −i Λp (s)Q1 (s) X ˜ ˜ θ e − θ u ¯ 1i 1 2i p + r1 A∗o (s) Λp (s)Q1 (s) Λp (s)Q1 (s) i=0 µ ¶ Λp (s)Q1 (s) ˜> αn+q−1 (s) > αn+q−1 (s) ˜ θ e − θ u ¯ 1 p + r1 (7.7.28) 1 2 A∗o (s) Λp (s)Q1 (s) Λp (s)Q1 (s) n ¯

Λp (s)Q1 (s) X ˙ ˙ r1 = Wci (s)[(Wbi (s)e1 ) θ˜1i − (Wbi (s)¯ up ) θ˜2i ] A∗o (s) i=0 4

From the definition of θ˜1 , we have θ˜1> αn+q−1 (s)

= θ1> αn+q−1 (s) − θ1∗> αn+q−1 (s) ˆ p (s, t)Qm (s) − sn+q − Rp (s)Qm (s) + sn+q = R ˆ p (s, t) − Rp (s))Qm (s) = θ˜a> αn−1 (s)Qm (s) = (R

(7.7.29)

4 where θ˜a = θa − θa∗ is the parameter error. Similarly,

θ˜2> αn+q−1 (s) = θ˜b> αn−1 (s)Q1 (s)

(7.7.30)

522

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

4 where θ˜b = θb − θb∗ . Using (7.7.29) and (7.7.30) in (7.7.28), we obtain µ ¶ Λp (s)Q1 (s) ˜> Qm (s) 1 1 > > ˜ C eo = θa αn−1 (s) e1 − θb αn−1 (s) u ¯ p + r1 A∗o (s) Q1 (s) Λp (s) Λp (s) ¶ µ Λp (s)Q1 (s) ˜> Qm (s) Qm (s) > ˜ = θa αn−1 (s) yp − θb αn−1 (s) up A∗o (s) Q1 (s)Λp (s) Λp (s)Q1 (s) +r1 (7.7.31)

where the second equality is obtained using Qm (s)e1 = Qm (s)yp , Noting that αn−1 (s)

1 up = φ1 , Λp (s)

Q1 (s)¯ up = Qm (s)up

αn−1 (s)

1 yp = −φ2 Λp (s)

we use Swapping Lemma A.1 to obtain the following equalities: ¢˙ ¡ Qm (s) ˜> Qm (s) ˜ θa φ2 = −θ˜a> αn−1 (s) yp + Wcq Wbq (s)φ> 2 θa Q1 (s) Q1 (s)Λp (s) ¡ ¢˙ Qm (s) Qm (s) ˜> ˜ θ φ1 = θ˜b> αn−1 (s) up + Wcq Wbq (s)φ> 1 θb Q1 (s) b Q1 (s)Λp (s) where Wcq , Wbq are as defined in Swapping Lemma A.1 with W (s) = the above equalities in (7.7.31) we obtain C > eo = −

Qm (s) Q1 (s) .

Λp (s)Qm (s) ˜> θp φ + r2 A∗o (s)

Using

(7.7.32)

where 4

r2 = r1 +

¡ ¢˙ ¡ ¢˙ ´ Λp (s)Q1 (s) ³ > ˜ > ˜ W (s) W (s)φ θ + W (s) W (s)φ θa cq bq b cq bq 1 2 A∗o (s)

From Table 7.5, the normalized estimation error satisfies the equation ²m2 = −θ˜p> φ which can be used in (7.7.32) to yield C > eo = 4

Λp (s)Qm (s) 2 ²m + r2 A∗o (s)

(7.7.33)

From the definition of m2f = 1 + kup k2 + kyp k2 and Lemma 3.3.2 , we can show that mf is a normalizing signal in the sense that φ/mf , m/mf ∈ L∞ for some

7.7. STABILITY PROOFS

523

δ > 0. From the expression of r1 , r2 and the normalizing properties of mf , we use Lemma 3.3.2 in (7.7.33) and obtain kC > eo k ≤ ck²mmf k + ckθ˙p mf k

(7.7.34)

Using (7.7.34) in (7.7.26) and in the definition of mf , we have the following inequality: m2f ≤ ck²mmf k2 + kθ˙p mf k2 + c or

m2f ≤ ckgmf k2 + c 4

where g 2 = ²2 m2 + |θ˙p |2 and g ∈ L2 , to which we can apply the B-G Lemma to show that mf ∈ L∞ . From mf ∈ L∞ and the properties of the L2δ norm we can establish boundedness for the rest of the signals. Step 4. Convergence of the tracking error to zero. The convergence of the tracking error to zero can be proved using the following arguments: Because all d (²m2 ) ∈ L∞ , which, together with signals are bounded, we can establish that dt 2 2 ²m ∈ L2 , implies that ²(t)m (t) → 0, and, therefore, θ˙p (t) → 0 as t → ∞. From the expressions of r1 , r2 we can conclude, using Corollary 3.3.1, that r2 ∈ L2 and r2 (t) → 0 as t → ∞. From (7.7.33), i.e., C > eo =

Λp (s)Qm (s) 2 ²m + r2 A∗o (s)

and ²m2 , r2 ∈ L2 and r2 → 0 as t → ∞, it follows from Corollary 3.3.1 that |C > eo | ∈ L2 and |C > eo (t)| → 0 as t → ∞. Because, from (7.7.23), we have eˆ˙ = Ac (t)ˆ e + Ko C > eo it follows from the u.a.s property of Ac (t) and the fact that |C > eo | ∈ L2 , |C > eo | → 0 4

as t → ∞ that eˆ → 0 as t → ∞. From e1 = yp − ym = C > eo + C > eˆ (see (7.7.24)), we conclude that e1 (t) → 0 as t → ∞ and the proof is complete.

7.7.3

Proof of Theorem 7.5.1

We present the stability proof for the hybrid scheme of Table 7.9. The proof of stability for the schemes in Tables 7.8 and 7.10 follows using very similar tools and arguments and is omitted. First, we establish the properties of the hybrid adaptive law using the following lemma. ˆ ok , K ˆ ck be as defined in Table 7.9 and θpk as defined Lemma 7.7.1 Let θ1k , θ2k , K in Table 7.7. The hybrid adaptive law of Table 7.7 guarantees the following:

524

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

(i) θpk ∈ `∞ and ∆θpk ∈ `2 , εm ∈ L2 ˆ ok ∈ `∞ and ∆θ1k , ∆θ2k ∈ `2 (ii) θ1k , θ2k , K 4

where ∆xk = xk+1 − xk ; k = 0, 1, 2, . . . ˆ p (s, tk )Qm (s), Zˆp (s, tk ) are strongly coprime for each k = 0, 1, . . ., then (iii) If R ˆ ck ∈ `∞ . K Proof The proof for (i) is given in Section 4.6. ˆ p (s, tk )Qm (s) − sn+q , By definition, θ1k , θ2k are the coefficient vectors of R ˆ p(s, tk ) = sn + θ> αn−1(s), Zˆp(s, tk ) = θ>αn−1(s). Zˆp (s, tk )Q1 (s) respectively, where R ak bk We can write θ1k = F1 θak , θ2k = F2 θbk where F1 , F2 ∈ R(n+q)×n are constant matrices which depend only on the coefficients of Q1 (s), Qm (s), respectively. Therefore, the properties θ1k , θ2k ∈ `∞ and ˆ ok = a∗ −θ1k , we have K ˆ ok ∈ `∞ ∆θ1k , ∆θ2k ∈ `2 follow directly from (i). Because K and the proof of (ii) is complete. Part (iii) is a direct result of linear system theory. 2 Lemma 7.5.1 indicates that the hybrid adaptive law given in Table 7.7 has essentially the same properties as its continuous counterpart. We use Lemma 7.7.1 to prove Theorem 7.5.1 for the adaptive law given in Table 7.9. As in the continuous time case, the proof is completed in four steps as follows: Step 1. Develop the state error equation for the closed-loop APPC scheme. Because the controller is unchanged for the hybrid adaptive scheme, we follow exactly the same steps as in the proof of Theorem 7.4.2 to obtain the error equations eˆ˙ = e˙ o = yp = up

=

ˆ ok (t)C > eo Ack (t)ˆ e+K Ao eo + θ˜1k (t)e1 − θ˜2k u ¯p > > C eo + C eˆ + ym ˆ ck (t)ˆ W1 (s)K e + W2 (s)yp

u ¯p

=

ˆ ck (t)ˆ −K e

(7.7.35)

ˆ ok , K ˆ ck , θ˜1k , θ˜2k are as defined in Section 7.4.3 with θb (t), θa (t) rewhere Ack , K ˆ ok = α∗ − θ1k and K ˆ ck placed by their discrete versions θbk , θak respectively, i.e., K ∗ ˆk K ˆ ck ) = Ac (s); Ao is a constant stable matrix, is solved from det(sI − Aˆk + B ˆk K ˆ ck and W1 (s), W2 (s) are proper stable transfer functions. FurtherAck = Aˆk − B ˆ ck (t), K ˆ ok (t), θ˜1k (t), θ˜2k (t) are piecewise constant functions defined more, Ack (t), K 4

as fk (t) = fk , ∀t ∈ [kTs , (k + 1)Ts ).

7.7. STABILITY PROOFS

525

Step 2. Establish e.s. for the homogeneous part of (7.7.35). Consider the system z˙ = A¯k z, z(t0 ) = z0 (7.7.36) where

· A¯k (t) =

ˆ ok (t)C > Ack (t) K 0 A0

¸ , t ∈ [kTs , (k + 1)Ts )

ˆk ) is strongly controllable, one can verify Because ∆θ1k , ∆θ2k ∈ `2 and (Aˆk , B ˆ ok , ∆K ˆ ck ∈ `2 and thus ∆Ack , ∆A¯k ∈ `2 . Therefore, for any given small that ∆K number µ > 0, we can find an integer Nµ such that2 kA¯k − A¯Nµ k < µ, ∀k ≥ Nµ We write (7.7.36) as z˙ = A¯Nµ z + (A¯k − A¯Nµ )z, ∀t ≥ Nµ Ts

(7.7.37)

Because A¯Nµ is a constant matrix that is stable and kA¯k − A¯Nµ k < µ, ∀k ≥ Nµ , we can fix µ to be small enough so that the matrix A¯k = A¯Nµ + (A¯k − A¯Nµ ) is stable which implies that |z(t)| ≤ c1 e−α(t−Nµ Ts ) |z(Nµ Ts )|, ∀t ≥ Nµ Ts for some constants c1 , α > 0. Because ¯

¯

¯

¯

z(Nµ Ts ) = eANµ Ts eANµ −1 Ts · · · eA1 Ts eA0 Ts z0 and Nµ is a fixed integer, we have |z(Nµ Ts )| ≤ c2 e−α(Nµ Ts −t0 ) |z0 | for some constant c2 > 0. Therefore, |z(t)| ≤ c1 c2 e−α(t−t0 ) |z0 |, ∀t ≥ Nµ Ts

(7.7.38)

where c1 , c2 , α are independent of t0 , z0 . On the other hand, for t ∈ [t0 , Nµ Ts ), we have ¯

2

¯

¯

¯

z(t) = eAi (t−iTs ) eAi−1 Ts · · · eA1 Ts eA0 Ts z0 P∞

(7.7.39)

If a sequence {fk } satisfies ∆fk ∈ `2 , then limn→∞ i=n k∆fi k = 0, which implies that for every given µ > 0, there exists an integer N °µ that depends ° on µ such that

Pk

i=Nµ

µ.

°Pk

k∆fi k < µ, ∀k ≥ Nµ . Therefore, kfk − fNµ k = °

i=Nµ

°

∆fi ° ≤

Pk

i=Nµ

k∆fi k <

526

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL

where i is the largest integer that satisfies i ≤ fixed integer, it follows from (7.7.39) that

t Ts

and i ≤ Nµ . Because Nµ is a

0

|z(t)| ≤ c3 |z0 | ≤ c3 e−α(t−t0 ) |z0 |, ∀t ∈ [t0 , Nµ Ts ) 0

4

(7.7.40)

0

for some constant c3 and c3 = c3 eαNµ Ts , which is independent of t0 , z0 . Combining (7.7.38) and (7.7.40), we have |z(t)| ≤ ce−α(t−t0 ) |z(t0 )|,

∀t ≥ t0

for some constants c, α > 0 which are independent of t0 , z0 and therefore (7.7.36), the homogeneous part of (7.7.35), is e.s. Step 3. Use the properties of the L2δ norm and B-G Lemma to establish signal boundedness. Following exactly the same procedure as in the proof of Theorem 7.4.2 presented in Section 7.7.2 for Step 3 of the continuous APPC, we can derive kmf k2 ≤ c + ckC > eo k2 where k · k denotes the L2δ norm, and C > eo = C > (sI − Ao )−1 (θ˜1k e1 − θ˜2k u ¯p )

(7.7.41)

Because ∆θ˜ak , ∆θ˜bk∈ l2 , according to Lemma 5.3.1, there exist vectors θ¯a(t), θ¯b (t) whose elements are continuous functions of time such that |θ˜ak (t) − θ¯a (t)| ∈ L2 , |θ˜bk (t) − θ¯b (t)| ∈ L2 and θ¯˙ a (t), θ¯˙ b (t) ∈ L2 . Let θ¯1 , θ¯2 be the vectors calculated from Table 7.9 by replacing θak , θbk with θ¯a , θ¯b respectively. As we have shown in the proof of Theorem 7.4.2, θ¯1 , θ¯2 depend linearly on θ¯a , θ¯b . Therefore, given the fact that θ¯˙ a , θ¯˙ b , |θ˜a − θ¯ak |, |θ˜b − θ¯bk | ∈ L2 , we have θ¯˙ 1 , θ¯˙ 2 , |θ˜1 − θ¯1k |, |θ˜2 − θ¯2k | ∈ L2 . Thus, we can write (7.7.41) as C > eo = C > (sI − Ao )−1 (θ¯1 e1 − θ¯2 u ¯p + e¯)

(7.7.42)

4 where e¯ = (θ˜1k (t) − θ¯1 (t))e1 − (θ˜2k (t) − θ¯2 (t))¯ up . Now, θ¯1 , θ¯2 have exactly the same ˜ ˜ properties as θ1 , θ2 in equation (7.7.27). Therefore, we can follow exactly the same procedure given in subsection 7.7.2 (from equation (7.7.27) to (7.7.33)) with the following minor changes: • Replace θ˜1 , θ˜2 by θ¯1 , θ¯2

• Replace θ˜a , θ˜b by θ¯a , θ¯b • Add an additional term C > (sI − Ao )−1 e¯ in all equations for C > eo • In the final stage of Step 3, to relate θ¯> φ2 − θ¯> φ1 with ²m2 , we write a

b

> > θ¯a> φ2 − θ¯b> φ1 = θ˜ak φ2 − θ˜bk φ1 + e¯2 4 where e¯2 = (θ¯a − θ˜ak )> φ2 − (θ¯b − θ˜bk )> φ1 , i.e., > > ²m2 = θ˜ak φ2 − θ˜bk φ1 = θ¯a> φ2 − θ¯b> φ1 − e¯2

7.8. PROBLEMS

527

At the end of this step, we derive the inequality m2f ≤ ck˜ g mf k2 + c 4 where g˜2 = ²2 m2 + |θ¯˙ p |2 + |θ˜pk − θ¯p |2 , θ¯p = [θ¯a> , θ¯b> ]> , and g˜ ∈ L2 . The rest of the proof is then identical to that of Theorem 7.4.2.

Step 4. Establish tracking error converges to zero. This is completed by following the same procedure as in the continuous case and is omitted here. 2

7.8

Problems

7.1 Consider the regulation problem for the first order plant x˙ = ax + bu y=x (a) Assume a and b are known and b 6= 0. Design a controller using pole placement such that the closed-loop pole is located at −5. (b) Repeat (a) using LQ control. Determine the value of λ in the cost function so that the closed-loop system has a pole at −5. (c) Repeat (a) when a is known but b is unknown. (d) Repeat (a) when b is known but a is unknown. 7.2 For the LQ control problem, the closed-loop poles can be determined from G(s) = Zp (s)/Rp (s) (the open-loop transfer function) and λ > 0 (the weighting of the plant input in the quadratic cost function) as follows: Define 4

F (s) = Rp (s)Rp (−s) + λ−1 Zp (s)Zp (−s). Because F (s) = F (−s), it can be factorized as F (s) = (s + p1 )(s + p2 ) · · · (s + pn )(−s + p1 )(−s + p2 ) · · · (−s + pn ) where pi > 0. Then, the closed-loop poles of the LQ control are equal to p1 , p2 , . . . , pn . (a) Using this property, give a procedure for designing an ALQC without having to solve a Riccati equation. (b) What are the advantages and disadvantages of this procedure compared with that of the standard ALQC described in Section 7.4.3? 7.3 Consider the speed control system given in Problem 6.2, where a, b, d are assumed unknown.

528

CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL (a) Design an APPC law to achieve the following performance specifications: (i) The time constant of the closed-loop system is less than 2 sec. (ii) The steady state error is zero for any constant disturbance d (b) Design an ALQC law such that Z J=

∞

(y 2 + λu2 )dt

0

is minimized. Simulate the closed-loop ALQC system with a = 0.02, b = 1.3, d = 0.5 for different values of λ. Comment on your results. 7.4 Consider the following system: y=

s2

ωn2 u + 2ξωn s + ωn2

where the parameter ωn (the natural frequency) is known, but the damping ratio ξ is unknown. The performance specifications for the closed-loop system are given in terms of the unit step response as follows: (a) the peak overshoot is less than 5% and (b) the settling time is less than 2 sec. (a) Design an estimation scheme to estimate ξ when ωn is known. (b) Design an indirect APPC and analyze the stability properties of the closed-loop system. 7.5 Consider the plant y=

s+b u s(s + a)

ˆ and ˆb, the estimate of a and b, (a) Design an adaptive law to generate a respectively, on-line. (b) Design an APPC scheme to stabilize the plant and regulate y to zero. ˆ and ˆb have to satisfy at each time (c) Discuss the stabilizability condition a t. (d) What additional assumptions you need to impose on the parameters a and b so that the adaptive algorithm can be modified to guarantee the stabilizability condition? Use these assumptions to propose a modified APPC scheme. 7.6 Repeat Problem 7.3 using a hybrid APPC scheme. 7.7 Solve the MRAC problem given by Problem 6.10 in Chapter 6 using Remark 7.3.1 and an APPC scheme. 7.8 Use Remark 7.3.1 to verify that the MRC law of Section 6.3.2 shown in Figure 6.1 is a special case of the general PPC law given by equations (7.3.3), (7.3.6).

7.8. PROBLEMS

529

7.9 Establish the stability properties of the hybrid ALQ control scheme given in Table 7.10. 7.10 Establish the stability properties of the hybrid APPC scheme of Table 7.8. 7.11 For n = 2, q = 1, show that A(t) defined in (7.7.5) satisfies det(sI − A(t)) = A∗ Λq , where A∗ , Λq are defined in Table 7.4.

Chapter 8

Robust Adaptive Laws 8.1

Introduction

The adaptive laws and control schemes developed and analyzed in Chapters 4 to 7 are based on a plant model that is free of noise, disturbances and unmodeled dynamics. These schemes are to be implemented on actual plants that most likely deviate from the plant models on which their design is based. An actual plant may be infinite dimensional, nonlinear and its measured input and output may be corrupted by noise and external disturbances. The effect of the discrepancies between the plant model and the actual plant on the stability and performance of the schemes of Chapters 4 to 7 may not be known until these schemes are implemented on the actual plant. In this chapter, we take an intermediate step, and examine the stability and robustness of the schemes of Chapters 4 to 7 when applied to more complex plant models that include a class of uncertainties and external disturbances that are likely to be present in the actual plant. The question of how well an adaptive scheme of the class developed in Chapters 4 to 7 performs in the presence of plant model uncertainties and bounded disturbances was raised in the late 1970s. It was shown, using simple examples, that an adaptive scheme designed for a disturbance free plant model may go unstable in the presence of small disturbances [48]. These examples demonstrated that the adaptive schemes of Chapters 4 to 7 are not robust with respect to external disturbances. This nonrobust behavior of adaptive schemes became a controversial issue in the early 1980s 530

8.2. PLANT UNCERTAINTIES AND ROBUST CONTROL

531

when more examples of instabilities were published demonstrating lack of robustness in the presence of unmodeled dynamics or bounded disturbances [85, 197]. This motivated many researchers to study the mechanisms of instabilities and find ways to counteract them. By the mid-1980s, several new designs and modifications were proposed and analyzed leading to a body of work known as robust adaptive control. The purpose of this chapter is to analyze various instability mechanisms that may arise when the schemes of Chapters 4 to 7 are applied to plant models with uncertainties and propose ways to counteract them. We start with Section 8.2 where we characterize various plant model uncertainties to be used for testing the stability and robustness of the schemes of Chapters 4 to 7. In Section 8.3, we analyze several instability mechanisms exhibited by the schemes of Chapters 4 to 7, in the presence of external disturbances or unmodeled dynamics. The understanding of these instability phenomena helps the reader understand the various modifications presented in the rest of the chapter. In Section 8.4, we use several techniques to modify the adaptive schemes of Section 8.3 and establish robustness with respect to bounded disturbances and unmodeled dynamics. The examples presented in Sections 8.3 and 8.4 demonstrate that the cause of the nonrobust behavior of the adaptive schemes is the adaptive law that makes the closed-loop plant nonlinear and time varying. The remaining sections are devoted to the development of adaptive laws that are robust with respect to a wide class of plant model uncertainties. We refer to them as robust adaptive laws. These robust adaptive laws are combined with control laws to generate robust adaptive control schemes in Chapter 9.

8.2

Plant Uncertainties and Robust Control

The first task of a control engineer in designing a control system is to obtain a mathematical model that describes the actual plant to be controlled. The actual plant, however, may be too complex and its dynamics may not be completely understood. Developing a mathematical model that describes accurately the physical behavior of the plant over an operating range is a challenging task. Even if a detailed mathematical model of the plant is available, such a model may be of high order leading to a complex controller whose implementation may be costly and whose operation may not be well

532

CHAPTER 8. ROBUST ADAPTIVE LAWS

understood. This makes the modeling task even more challenging because the mathematical model of the plant is required to describe accurately the plant as well as be simple enough from the control design point of view. While a simple model leads to a simpler control design, such a design must possess a sufficient degree of robustness with respect to the unmodeled plant characteristics. To study and improve the robustness properties of control designs, we need a characterization of the types of plant uncertainties that are likely to be encountered in practice. Once the plant uncertainties are characterized in some mathematical form, they can be used to analyze the stability and performance properties of controllers designed using simplified plant models but applied to plants with uncertainties.

8.2.1

Unstructured Uncertainties

Let us start with an example of the frequency response of a stable plant. Such a response can be obtained in the form of a Bode diagram by exciting the plant with a sinusoidal input at various frequencies and measuring its steady state output response. A typical frequency response of an actual stable plant with an output y may have the form shown in Figure 8.1. It is clear that the data obtained for ω ≥ ωm are unreliable because at high frequencies the measurements are corrupted by noise, unmodeled high frequency dynamics, etc. For frequencies below ωm , the data are accurate enough to be used for approximating the plant by a finite-order model. An approximate model for the plant, whose frequency response is shown in Figure 8.1, is a second-order transfer function G0 (s) with one stable zero and two poles, which disregards the phenomena beyond, say ω ≥ ωm . The modeling error resulting from inaccuracies in the zero-pole locations and high frequency phenomena can be characterized by an upper bound in the frequency domain. Now let us use the above example to motivate the following relationships between the actual transfer function of the plant denoted by G(s) and the transfer function of the nominal or modeled part of the plant denoted by G0 (s). Definition 8.2.1 (Additive Perturbations) Suppose that G(s) and G0 (s) are related by G(s) = G0 (s) + ∆a (s) (8.2.1)

8.2. PLANT UNCERTAINTIES AND ROBUST CONTROL

533

5

0

-5

20 log

10

y ( jω ) -10

-15

-20 -1 10

0

10

1

10

2

ωm

10

ω

Figure 8.1 An example of a frequency response of a stable plant.

where ∆a (s) is stable. Then ∆a (s) is called an additive plant perturbation or uncertainty. The structure of ∆a (s) is usually unknown but ∆a (s) is assumed to satisfy an upper bound in the frequency domain, i.e., |∆a (jω)| ≤ δa (ω) ∀ω

(8.2.2)

for some known function δa (ω). In view of (8.2.1) and (8.2.2) defines a family of plants described by Πa = {G | |G(jω) − G0 (jω)| ≤ δa (ω) }

(8.2.3)

The upper bound δa (ω) of ∆a (jω) may be obtained from frequency response experiments. In robust control [231], G0 (s) is known exactly and the

534

CHAPTER 8. ROBUST ADAPTIVE LAWS

uncertainties of the zeros and poles of G(s) are included in ∆a (s). In adaptive control, the parameters of G0 (s) are unknown and therefore zero-pole inaccuracies do not have to be included in ∆a (s). Because the main topic of the book is adaptive control, we adopt Definition 8.2.1, which requires ∆a (s) to be stable. Definition 8.2.2 (Multiplicative Perturbations) Let G(s), G0 (s) be related by G(s) = G0 (s)(1 + ∆m (s)) (8.2.4) where ∆m (s) is stable. Then ∆m (s) is called a multiplicative plant perturbation or uncertainty. In the case of multiplicative plant perturbations, ∆m (s) may be constrained to satisfy an upper bound in the frequency domain, i.e., |∆m (jω)| ≤ δm (ω)

(8.2.5)

for some known δm (ω) which may be generated from frequency response experiments. Equations (8.2.4) and (8.2.5) describe a family of plants given by ½ ¯ ¾ ¯ |G(jω) − G0 (jω)| Πm = G ¯¯ ≤ δm (ω) (8.2.6) |G0 (jω)| For the same reason as in the additive perturbation case, we adopt Definition 8.2.2 which requires ∆m (s) to be stable instead of the usual definition in robust control where ∆m (s) is allowed to be unstable for a certain family of plants. Definition 8.2.3 (Stable Factor Perturbations) Let G(s), G0 (s) have the following coprime factorizations [231]: G(s) =

N0 (s) + ∆1 (s) N0 (s) , G0 (s) = D0 (s) + ∆2 (s) D0 (s)

(8.2.7)

where N0 and D0 are proper stable rational transfer functions that are coprime,1 and ∆1 (s) and ∆2 (s) are stable. Then ∆1 (s) and ∆2 (s) are called stable factor plant perturbations. 1 Two proper transfer functions P (s), Q(s) are coprime if and only if they have no finite common zeros in the closed right half s-plane and at least one of them has relative degree zero [231].

8.2. PLANT UNCERTAINTIES AND ROBUST CONTROL

535

- ∆a (s)

u

+ + ? Σl

- G0 (s)

-y

(i) - ∆m (s)

+ + ? Σl - G0 (s)

u

y-

(ii)

- ∆1 (s)

u

- N0 (s)

+ ?+ - ΣlΣl +

−6

y+ ¾ Σl + 6

D0 (s)−1 ¾ ∆2 (s) ¾

(iii) Figure 8.2 Block diagram representations of plant models with (i) additive, (ii) multiplicative, and (iii) stable factor perturbations.

Figure 8.2 shows a block diagram representation of the three types of plant model uncertainties. The perturbations ∆a (s), ∆m (s), ∆1 (s), and ∆2 (s) defined above with no additional restrictions are usually referred to as unstructured plant model uncertainties.

536

8.2.2

CHAPTER 8. ROBUST ADAPTIVE LAWS

Structured Uncertainties: Singular Perturbations

In many applications, the plant perturbations may have a special form because they may originate from variations of physical parameters or arise because of a deliberate reduction in the complexity of a higher order mathematical model of the plant. Such perturbations are usually referred to as structured plant model perturbations. The knowledge of the structure of plant model uncertainties can be exploited in many control problems to achieve better performance and obtain less conservative results. An important class of structured plant model perturbations that describe a wide class of plant dynamic uncertainties, such as fast sensor and actuator dynamics, is given by singular perturbation models [106]. For a SISO, LTI plant, the following singular perturbation model in the state space form x˙ = A11 x + A12 z + B1 u,

x ∈ Rn

µz˙ = A21 x + A22 z + B2 u,

z ∈ Rm

(8.2.8)

y = C1> x + C2> z can be used to describe the slow (or dominant) and fast (or parasitic) phenomena of the plant. The scalar µ represents all the small parameters such as small time constants, small masses, etc., to be neglected. In most applications, the representation (8.2.8) with a single parameter µ can be achieved by proper scaling as shown in [106]. All the matrices in (8.2.8) are assumed to be constant and independent of µ. As explained in [106], this assumption is for convenience only and leads to a minor loss of generality. The two time scale property of (8.2.8) is evident if we use the change of variables zf = z + L(µ)x (8.2.9) where L(µ) is required to satisfy the algebraic equation A21 − A22 L + µLA11 − µLA12 L = 0

(8.2.10)

to transform (8.2.8) into x˙ = As x + A12 zf + B1 u µz˙f = Af zf + Bs u y = Cs> x + C2> zf

(8.2.11)

8.2. PLANT UNCERTAINTIES AND ROBUST CONTROL

537

where As = A11 −A12 L, Af = A22 +µLA12 , Bs = B2 +µLB1 , Cs> = C1>− C2> L. As shown in [106], if A22 is nonsingular, then for all µ ∈ [0, µ∗ ) and some µ∗ > 0, a solution of the form L = A−1 22 A21 + O(µ) satisfying (8.2.10) exists. It is clear that for u = 0, i.e., x˙ = As x + A12 zf µz˙f = Af zf

(8.2.12)

the eigenvalues of (8.2.12) are equal to those of As and Af /µ, which, for small µ and for Af nonsingular, are of O(1) and O(1/µ), respectively2 . The smaller the value of µ, the wider the distance between the eigenvalues of As and Af /µ, and the greater the separation of time scales. It is clear that if Af is stable then the smaller the value of µ is, the faster the state variable zf goes to zero. In the limit as µ → 0, zf converges instantaneously, i.e., infinitely fast to zero. Thus for small µ, the effect of stable fast dynamics is reduced considerably after a very short time. Therefore, when A22 is stable (which for small µ implies that Af is stable), a reasonable approximation of (8.2.8) is obtained by setting µ = 0, solving for z from the second equation of (8.2.8) and substituting for its value in the first equation of (8.2.8), i.e., x˙ 0 = A0 x0 + B0 u, x0 ∈ Rn y0 = C0> x0 + D0 u

(8.2.13)

−1 > > > −1 where A0 = A11 − A12 A−1 22 A21 , B0 = B1 − A12 A22 B2 , C0 = C1 − C2 A22 A21 and D0 = −C2> A−1 22 B2 . With µ set to zero, the dimension of the state space of (8.2.8) reduces from n + m to n because the differential equation for z in (8.2.8) degenerates into the algebraic equation

0 = A21 x0 + A22 z0 + B2 u i.e., z0 = −A−1 22 (A21 x0 + B2 u) 2

(8.2.14)

We say that a function f (x) is O(|x|) in D ⊂ Rn if there exists a finite constant c > 0 such that |f (x)| ≤ c|x| for all x ∈ D where x = 0 ∈ D.

538

CHAPTER 8. ROBUST ADAPTIVE LAWS

where the subscript 0 is used to indicate that the variables belong to the system with µ = 0. The transfer function G0 (s) = C0> (sI − A0 )−1 B0 + D0

(8.2.15)

represents the nominal or slow or dominant part of the plant. We should emphasize that even though the transfer function G(s) from u to y of the full-order plant given by (8.2.8) is strictly proper, the nominal part G0 (s) may be biproper because D0 = −C2> A−1 22 B2 may not be equal to zero. The situation where the throughput D0 = −C2> A−1 22 B2 induced by the fast dynamics is nonzero is referred to as strongly observable parasitics [106]. As discussed in [85, 101], if G0 (s) is assumed to be equal to C0> (sI − A0 )−1 B0 instead of (8.2.15), the control design based on G0 (s) and applied to the full-order plant with µ ≥ 0 may lead to instability. One way to eliminate the effect of strongly controllable and strongly observable parasitics is to augment (8.2.8) with a low pass filter as follows: We pass y through the f1 filter s+f for some f1 , f0 > 0, i.e., 0 y˙ f = −f0 yf + f1 y

(8.2.16)

and augment (8.2.8) with (8.2.16) to obtain the system of order (n + 1 + m) ˆ1 u x˙ a = Aˆ11 xa + Aˆ12 z + B ˆ µz˙ = A21 xa + A22 z + B2 u yˆ = Cˆ1> xa

(8.2.17)

ˆ1 , Cˆ1 are appropriately defined. where xa = [yf , x> ]> and Aˆ11 , Aˆ12 , Aˆ21 , B The nominal transfer function of (8.2.17) is now ˆ 0 (s) = Cˆ0> (sI − Aˆ0 )−1 B ˆ0 G which is strictly proper. Another convenient representation of (8.2.8) is obtained by using the change of variables η = zf + A−1 (8.2.18) f Bs u i.e., the new state η represents the difference between the state zf and the “quasi steady” state response of (8.2.11) due to µ 6= 0 obtained by approxi-

8.2. PLANT UNCERTAINTIES AND ROBUST CONTROL

539

mating µz˙f ≈ 0. Using (8.2.18), we obtain ¯s u x˙ = As x + A12 η + B −1 µη˙ = Af η + µAf Bs u˙ y = Cs> x + C2> η + Ds u

(8.2.19)

¯s = B1 − A12 A−1 Bs , Ds = −C > A−1 Bs , provided u is differentiable. where B 2 f f Because for |u| ˙ = O(1), the slow component of η is of O(µ), i.e., at steady state |η| = O(µ), the state η is referred to as the parasitic state. It is clear that for |u| ˙ = O(1) the effect of η on x at steady state is negligible for small µ whereas for |u| ˙ ≥ O(1/µ), |η| is of O(1) at steady state, and its effect on the slow state x may be significant. The effect of u˙ and η on x is examined in later sections in the context of robustness of adaptive systems.

8.2.3

Examples of Uncertainty Representations

We illustrate various types of plant uncertainties by the following examples: Example 8.2.1 Consider the following equations describing the dynamics of a DC motor J ω˙ = k1 i di L = −k2 ω − Ri + v dt where i, v, R and L are the armature current, voltage, resistance, and inductance, respectively; J is the moment of inertia; ω is the angular speed; k1 i and k2 ω are the torque and back e.m.f., respectively. Defining x = ω, z = i we have x˙ = b0 z, µz˙ = −α2 x − α1 z + v y=x where b0 = k1 /J, α2 = k2 , α1 = R and µ = L, which is in the form of the singular perturbation model (8.2.8). The transfer function between the input v and the output y is given by y(s) b0 = 2 = G(s, µs) v(s) µs + α1 s + α0 where α0 = b0 α2 . In most DC motors, the inductance L = µ is small and can be neglected leading to the reduced order or nominal plant transfer function G0 (s) =

b0 α1 s + α0

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CHAPTER 8. ROBUST ADAPTIVE LAWS

Using G0 (s) as the nominal transfer function, we can express G(s, µs) as G(s, µs) = G0 (s) + ∆a (s, µs), ∆a (s, µs) = −µ

b0 s2 (µs2 + α1 s + α0 )(α1 s + α0 )

where ∆a (s, µs) is strictly proper and stable since µ, α1 , α0 > 0 or as G(s, µs) = G0 (s)(1 + ∆m (s, µs)), ∆m (s, µs) = −µ

s2 µs2 + α1 s + α0

where ∆m (s, µs) is proper and stable. Let us now use Definition 8.2.3 and express G(s, µs) in the form of stable factor perturbations. We write G(s, µs) as G(s, µs) =

b0 (s+λ) + ∆1 (s, µs) α1 s+α0 (s+λ) + ∆2 (s, µs)

where ∆1 (s, µs) = −µ

b0 s α0 s , ∆2 (s, µs) = −µ (µs + α1 )(s + λ) (µs + α1 )(s + λ)

and λ > 0 is an arbitrary constant.

5

Example 8.2.2 Consider a system with the transfer function G(s) = e−τ s

1 s2

where τ > 0 is a small constant. As a first approximation, we can set τ = 0 and obtain the reduced order or nominal plant transfer function G0 (s) =

1 s2

leading to G(s) = G0 (s)(1 + ∆m (s)) −τ s

with ∆m (s) = e

− 1. Using Definition 8.2.3, we can express G(s) as G(s) = 2

N0 (s) + ∆1 (s) D0 (s) + ∆2 (s) −τ s

1 s e −1 where N0 (s) = (s+λ) 2 , D0 (s) = (s+λ)2 , ∆1 (s) = (s+λ)2 , and ∆2 (s) = 0 where λ > 0 is an arbitrary constant. It is clear that for small τ , ∆m (s), ∆1 (s) are approximately equal to zero at low frequencies. 5

8.2. PLANT UNCERTAINTIES AND ROBUST CONTROL du ym + leC(s) Σ − 6

ye

d

0 u-+ ? luG(s) +Σ

F (s) ¾

541

yn

+ ? - Σl y+ +? d ¾ n Σl +

Figure 8.3 General feedback system.

8.2.4

Robust Control

The ultimate goal of any control design is to meet the performance requirements when implemented on the actual plant. In order to meet such a goal, the controller has to be designed to be insensitive, i.e., robust with respect to the class of plant uncertainties that are likely to be encountered in real life. In other words, the robust controller should guarantee closed-loop stability and acceptable performance not only for the nominal plant model but also for a family of plants, which, most likely, include the actual plant. Let us consider the feedback system of Figure 8.3 where C, F are designed to stabilize the nominal part of the plant model whose transfer function is G0 (s). The transfer function of the actual plant is G(s) and du , d, dn , ym are external bounded inputs as explained in Section 3.6. The difference between G(s) and G0 (s) is the plant uncertainty that can be any one of the forms described in the previous section. Thus G(s) may represent a family of plants with the same nominal transfer function G0 (s) and plant uncertainty characterized by some upper bound in the frequency domain. We say that the controller (C, F ) is robust with respect to the plant uncertainties in G(s) if, in addition to G0 (s), it also stabilizes G(s). The property of C, F to stabilize G(s) is referred to as robust stability. The following theorem defines the class of plant uncertainties for which the controller C, F guarantees robust stability. Theorem 8.2.1 Let us consider the feedback system of Figure 8.3 where (i) G(s) = G0 (s) + ∆a (s) (ii) G(s) = G0 (s)(1 + ∆m (s))

542

CHAPTER 8. ROBUST ADAPTIVE LAWS N0 (s) + ∆1 (s) N0 (s) , G0 (s) = D0 (s) + ∆2 (s) D0 (s)

(iii) G(s) =

where ∆a (s), ∆m (s), ∆1 (s), ∆2 (s), N0 (s), and D0 (s) are as defined in Section 8.2.1 and assume that C, F are designed to internally stabilize the feedback system when G(s) = G0 (s). Then the feedback system with G(s) given by (i), (ii), (iii) is internally stable provided conditions (i)

(ii)

° ° ° ° C(s)F (s) ° ° ° 1 + C(s)F (s)G (s) ° δa (ω) < 1 0 ∞ ° ° ° C(s)F (s)G0 (s) ° ° ° ° 1 + C(s)F (s)G (s) ° δm (ω) < 1 0

(8.2.21)

∞

° ° ° ∆2 (s) + C(s)F (s)∆1 (s) ° ° ° (iii) ° D (s) + C(s)F (s)N (s) ° 0

(8.2.20)

0

<1

(8.2.22)

∞

are satisfied for all ω, respectively. Proof As in Sections 3.6.1 and 3.6.2, the feedback system is internally stable if and only if each element of the transfer matrix   1 −F G −F −F  C 1 1 −F C −F C    H(s) = G 1 −F CG  1 + F CG  CG CG G 1 1 has stable poles. Because C, F are designed such that each element of H(s) with G = G0 has stable poles and ∆α , ∆m , ∆1 , and ∆2 are assumed to have stable poles, the only instability which may arise in the feedback system is from any unstable root of 1 + F (s)C(s)G(s) = 0 (8.2.23) (i) Let us consider G = G0 + ∆a , then (8.2.23) can be written as 1+

F C∆a =0 1 + F CG0

(8.2.24)

FC Because 1+F CG0 , ∆a have stable poles, it follows from the Nyquist criterion that F C∆a the roots of (8.2.24) are in the open left-half s-plane if the Nyquist plot of 1+F CG0 does not encircle the (−1, j0) point in the complex s = σ +jω plane. This condition is satisfied if ¯ ¯ ¯ C(jω)F (jω)δa (ω) ¯ ¯ ¯ ¯ 1 + C(jω)F (jω)G0 (jω) ¯ < 1, ∀ω ∈ R

8.2. PLANT UNCERTAINTIES AND ROBUST CONTROL

543

F C∆a The above condition implies that the Nyquist plot of 1+F CG0 does not encircle the (−1, j0) point for any ∆a (s) that satisfies |∆a (jω)| ≤ δa (ω). Hence, the feedback system is internally stable for any G(s) in the family

Πa = {G | |G(jω) − G0 (jω)| ≤ δa (ω) } (ii) If G = G0 (1 + ∆m ) equation (8.2.23) may be written as 1+

F CG0 ∆m = 0 1 + F CG0

and the proof follows as in (i). (iii) If G =

N0 +∆1 D0 +∆2 ,

equation (8.2.23) may be written as 1+

1 [∆2 + F C∆1 ] = 0 (D0 + F CN0 )

FC Because D0 +F1 CN0 , D0 +F CN0 , ∆2 , ∆1 have stable poles, the result follows by applying the Nyquist criterion as in (i). 2

We should emphasize that conditions (8.2.20) to (8.2.22) are not only sufficient for stability but also necessary in the sense that if they are violated, then within the family of plants considered, there exists a plant G for which the feedback system with compensators F, C is unstable. Conditions (8.2.20) to (8.2.22) are referred to as conditions for robust stability. They may be used to choose F, C such that in addition to achieving internal stability for the nominal plant, they also guarantee robust stability with respect to a class of plant uncertainties. As in Section 3.6.3, let us consider the performance of the feedback system of Figure 8.3 with respect to the external inputs in the presence of dynamic plant uncertainties. We concentrate on the case where F (s) = 1 and dn = du = 0 and the plant transfer function is given by G(s) = G0 (s)(1 + ∆m (s))

(8.2.25)

By performance, we mean that the plant output y is as close to the reference input ym as possible for all plants G in the family Πm despite the presence of the external inputs ym , d. From Figure 8.3, we have y=

1 CG ym + d 1 + CG 1 + CG

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CHAPTER 8. ROBUST ADAPTIVE LAWS

or e = ym − y =

1 (ym − d) 1 + CG

(8.2.26)

Because 1 + CG = 1 + CG0 + CG0 ∆m , (8.2.26) can be expressed as e=

S0 (ym − d) 1 + T0 ∆m

(8.2.27)

CG0 1 where S0 = 1+CG , T0 = 1+CG are the sensitivity and complementary sen0 0 sitivity functions for the nominal plant, respectively. For robust stability we require

° ° ° C(s)G0 (s) ° ° ° kT0 (s)k∞ δm (ω) = ° 1 + C(s)G (s) ° 0

δm (ω) < 1

∞

which suggests that the loop gain L0 = CG0 should be much less than 1 whenever δm (ω) exceeds 1. For good tracking performance, i.e., for small 1 error e, however, (8.2.27) implies that S0 = 1+CG should be small, which, in 0 turn, implies that the loop gain L0 should be large. This is the classical tradeoff between nominal performance and robust stability that is well known in the area of robust control. A good compromise may be found when δm (ω) < 1 at low frequencies and v = ym − d is small at high frequencies. In this case the loop gain L0 = CG0 can be shaped, through the cho